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SPRING ENGINEERING 


A TEXT-BOOK 

FOR ENGINEERS, STUDENTS, AND 
DRAUGHTSMEN 


— by ^4 

EGBERT R^MORRISON 

JUN. AM. SOC. M. E. 


PRICE, $4.00 

Delivered in the United States 


Published and for sale by 

EGBERT R. MORRISON 
SHARON, PA., U.S.A. 
1915 





Copyright, 1915 
by 

EGBERT R. MORRISON 
Sharon, Pa. 



JAN 21 1916 


©CI.A420452 


"H-t? / , 

y ! 













■ 


PREFACE 


Springs are among the most ancient of machinery ele¬ 
ments; yet because they have lent themselves more or less 
readily to experimental designing, and because a large 
amount of work is involved in arriving at their proper di¬ 
mensions, little investigation has been done along purely sci¬ 
entific lines. Such investigation has been further retarded 
by a prevalent assumption that a spring is an element of 
secondary importance. Many mechanisms have failed to 
give the results and service expected because of a failure to 
provide springs carefully fitted to the work to be performed. 

The following articles have been written largely in 
answer to the numerous inquiries which followed the publi¬ 
cation of our “Spring Tables.” It is hoped that they will 
be of assistance in connection with the better known or 
more common forms of springs. The work is not intended 
to cover the field thoroughly and completely, as the subject 
is one which can scarcely be exhausted. 

Such tables as are introduced are taken largely from 
“Spring Tables,” and used here on account of their value in 
connection with articles along the same lines. 

For assistance in developing this book I have to 
acknowledge my indebtedness: to “Machinery,” for per¬ 
mission to reprint articles originally published by them; to 
The Wm. D. Gibson Co., for many valuable suggestions; 
and to Mr. Clarence K. Sheers, for assistance in preparing 
the tabular work. 


Egbert R. Morrison 


December 1, 1915. 

































- 






. 







































, 

















‘ JL. 













CONTENTS 

I. Fundamental Principles. 7 

II. Elliptic or Leaf Springs. 17 

III. Spiral Springs. 28 

IV. Heavy Helical Springs.30 

V. Grouped Helical Springs. 37 

VI. Conical Helical Springs. 49 

VII. Wire Springs.\. 59 

VIII. Elliptical and Rectangular Sections. 66 

IX. Mathematical Tables. 73 

X. Advertisements.. 76 













Various Types of Helical Springs 



















CHAPTER I. 


FUNDAMENTAL PRINCIPLES OF 
SPRING DESIGNING 


The fundamental principles of spring designing may be discussed 
under the headings: 

1. Elasticity 

2. Physical characteristics 

3. Properties of shape 

4. Bending formulas 

5. Torsion formulas 

6. Classification of springs 

7. Adaptation of type to work 

8. Workmanship 


Elasticity 

If we bend or twist a hickory stick we find that it opposes our 
action, and that, if we have not bent or twisted it so far as to destroy 
its original nature, it will, whether bent or twisted, return to its orig¬ 
inal shape. All other materials also possess this quality to a greater 
or less extent. It has been noticed that both the material and shape 
of the stick have much to do with the extent to which same can be 
safely distorted, this quality of elasticity, as it is called, depending 
upon these two properties. Engineers have, therefore, by testing 
similar shapes made of different materials, and different shapes made 
of the same material, been able to come to an understanding of the 
physical properties of matter and the laws of shape in relation to elas¬ 
ticity. 

Since the return of a distorted body to its original shape is, 
in popular language, a “springing” back to original shape, such bodies 
have come to be known as springs. 


8 


SPRING ENGINEERING. 


Physical Characteristics 

A spring is affected by the material of which it is made, because 
two of the physical characteristics of matter enter into the laws of 
elasticity. Thus, the safe stress which may be excited within any 
material depends entirely upon the nature of that material. Again, 
the extent to which a material can be distorted without resulting in a 
permanent distortion depends entirely upon the material. 

The safe maximum stress is known for different materials solely 
by experiment and in mathematical discussion is generally repre¬ 
sented by S. 

The index to the extent of safe distortion is also found by direct 
experiments, and is called the “modulus of elasticity.” Since it is 
possible to distort materials in different ways there are different 
“moduli of elasticity.” Thus we have, 

1. The modulus of elasticity for tension. 

2. The modulus of elasticity for compression. 

3. The modulus of elasticity for shear. 

4. The modulus of elasticity for bending. 

5. The modulus of elasticity for twisting. 

While springs could, undoubtedly, be designed based upon the 
action of any of these forces, yet the engineer is concerned practically 
with only the two classes of springs: 

1. Those involving a bending action. 

2. Those involving a twisting action. 

It is interesting to note that neither of these classes is the result 
of a simple stress. Bending is a combination of tension and compres¬ 
sion. In its application part of the fibres are in tension, part in 
compression, and between the two there is a neutral line known as 
“the neutral axis”—neutral to both tension and compression. Twist¬ 
ing may be defined as a special shearing action so applied that the 
resultant shearing stress in any fibre depends upon its distance from 
the neutral axis or center line of twist—i.e., the stress diminishes 
upon approaching the mechanical center, until it is zero or neutral 
at that center. 

It is quite possible, however, that springs based upon the action 
of direct tension, compression, and shear will in the future demand 
more thorough investigation in connection with the use of materials 
like rubber. 

The modulus of elasticity for bending is not usually distinguish¬ 
able from that of tension, so that the effect of the compression pres¬ 
ent is overlooked. More careful investigation may in the future be 
very profitable in establishing for different materials the correct mod¬ 
ulus of elasticity for bending. The modulus of elasticity both for ten¬ 
sion and bending is now generally expressed by E. 



SPRING ENGINEERING. 


9 


The “modulus of elasticity” is a term used to express the ratio 
between the applied load and the resultant distortion. Thus 

Force of tension 

E =-.- 

Resultant extension per unit of length 

By custom, unless otherwise stated, the use of the term “modu¬ 
lus of elasticity” relates to the modulus for tension, or E. 

In speaking of the other moduli it is usual to say “/or compres¬ 
sion ,” “/or torsion ,” etc. Thus, the modulus of elasticity for torsion 
is usually expressed by G. 

The modulus of the torsional elasticity is affected by the fact that 
the maximum stress acts in an oblique plane and is 25 per cent greater 
than the stress in the normal plane at any point. Therefore, the 
maximum value of S must be taken as only ^ of a safe, true maxi¬ 
mum stress. In the same way the modulus of elasticity for torsion is 
not V 2 E but Vz (y 5 E) = % E = G. 

Values of E for 
Various Spring Materials 


Steel, cast, spring tempered, high carbon. 40,000,000 

Steel, P. R. R. analysis spring, tempered..30,000,000 

Steel, machinery .30,000,000 

Steel, soft .28,000,000 

Wrought iron .25,000,000 

Platinum .24,000,000 

Copper wire .18,500,000 

Cast iron ....18,000,000 

Copper, hammered .15,600,000 

Copper, cast . 15,000,000 

Brass, wire .14,200,000 

Bronze, phosphor .14,000,000 

Gold .11,500,000 

Bronze, gun metal .10,000,000 

Aluminum . 9,000,000 

Lead . 2,500,000 


The strength of materials is greatly increased by the process of 
drawing, for which reason the strength of wire exceeds the strength 
of other forms such as rods, sheets, and castings. The tables given 
are for direct tension. What might be called the torsional stresses 
are j4 of these true stresses. 

A large manufacturer of brass wire states that brass has a ten¬ 
dency towards brittleness and should, therefore, not be used for se¬ 
vere service. Two-and-one brass is largely used, which, while not so 
strong as common brass, will withstand more severe service. The 
maximum toughness for brass is reached in the proportions of 72 
per cent copper and 28 per cent zinc. 





















10 


SPRING ENGINEERING. 


Gun metal makes a stronger spring but not quite as long lived a 
spring as phosphor bronze, which is used very largely where repeated 
action is taking place, as in electrical fixtures. 

Brass and bronze wires are liable to become more or less brittle 
after long storage, so that springs should be made up as soon as 
possible, the use of the spring seeming to break up the continuity of 
strains and tendency to crystallization. 

Ultimate Resistance to Tension of Bars, Sheets, Castings 
Pounds Per Square Inch 

Steel .45,000 to 120,000 

Steel, aluminum 2y$ per cent.70,000 

Steel, copper 35 per cent.60,000 

Steel, nickel 3% per cent. 86,000 

Iron cast .13,400 to 29,000 

Iron, wrought .55,000 

Copper, cast .19,000 

Copper, sheets .30,000 

Copper, bolts .36,000 

Brass, cast .18,000 

Aluminum Bronze 

10 per cent Al. 90 per cent Cu.85,000 

1% per cent Al. 98% per cent Cu.28,000 

Aluminum 

Castings .12,000 to 14,000 

Sheet .24,000 to 40,000 

Bars .28,000 to 40,000 

Bronze, gun metal.36,000 

Lead, sheet. 3,300 

Ultimate Resistance of Tension of Wire 
Pounds Per Square Inch 

Steel, cast, crucible .224,000 

Steel, Bessemer . 86,600 

Steel, high carbon.179,200 

Steel, mild O. H.134,000 

Iron, black or annealed. 56,000 

Iron, bright hard drawn . 78,400 

Copper, unannealed . 60,000 

Brass . 49,000 to 90,000 

Brass, spring tempered. 90,000 to 100,000 

Bronze, phosphor .130,000 to 140,000 

Bronze, gun metal . 140,000 to 150,000 

Aluminum . 25,000 to 55,000 
































SPRING ENGINEERING. 


11 


Properties of Shape 

The question of shape enters also into spring designing because 
the shape of a bar decides two things which are controlling factors 
even as much as the two physical properties of the material. Thus, 
as we had under physical properties, <S and E (or G) so we have un¬ 
der laws of shape c and I (or J). 

Upon shape depends the distance of the remotest fibre from the 
neutral axis. This distance is usually denoted by c and the remotest 
fibre is the one which by the leverage action of bending or twisting, 
will be most highly stressed. 

Upon shape depends also the distribution, about the neutral axis, 
of the elementary areas in any cross section, and upon this distribu¬ 
tion depends I and J , the rectangular and polar moments of inertia. 
The load which may be safely applied to distort without permanent 
distortion, varies directly as the value of these moments of inertia. 

The rectangular moment of inertia, 7, has to do with bending. 
The polar moment, G, has to do with twisting. 


Fundamental Bending Formulas—Load 

When a bar is subject to a bending action there is present an 
acting moment tending to produce this bending moment and a resist¬ 
ing moment which holds the acting moment in equilibrium. 

The acting moment is due to the force P acting at the end of the 
leverage l and is therefore expressed by PI. 

The resisting moment is expressed by the product of stress times 
rectangular moment of inertia divided by the distance from the neu¬ 
tral axis, or SI 

c 

Then, 

Twisting moment = Resisting moment 
SI 

Pl = - 

c 

SI 

P — - 

cl 

bh 3 h 

Or, since 7 =- and c = — for rectangular shapes, 

12 2 

Sbh* 

P = -, the fundamental formula for load. 


6 










12 


SPRING ENGINEERING . 


Fundamental Bending Formula—Deflection 

The general equation of the elastic curve applicable to all beams 
whatever be their shapes, loads, or number of spans is 

d 2 y M 


dx 2 El 


And since M — Px 


b <P 

And I — - 

12 


d 2 y 12 P x 

- == - for uniform sections. (A) 

dx 2 Ebd? 

In a cantilever of uniform strength and depth 
6 P l 

b = - 

Sd? 

.'.<?y 2 S 

-=-, for uniform strength and depth. (B) 

dx 2 Ed 

The double integration of these two equations (A and B) results 
in the expressions for the elastic curves, and the substitution of x = l 
gives the maximum deflections, thus: 

4PP 


/ = 


-, uniform sections. 

Ebd 3 



6 P P 


/ = 


-, uniform strength and depth. 


Ebd 3 


Which are the fundamental formulas for deflection. 


Torsion Action of Helical Springs 

The same load applied through a radius R, equivalent to the mean 
radius of helix, will, when applied to a simple torsion spring, produce 
very nearly the same movement of the load, or the same deflection. 
As stated by Mr. Oberlin Smith, after making over 200 experiments: 

“to ascertain the distance which one end of a proposed spring 
will move with a given weight attached, it is only necessary to take 
a straight wire of the same diameter, length, and material, and, fix¬ 
ing one end, twist the other end with the same weight hung from the 
periphery of a wheel the same size as the mean diameter of the pro¬ 
posed coils—noting, of course, the distance moved by the weight.” 

There is also a bending action in helical springs but so slight 
that it is neglected in practical considerations. 












SPRING ENGINEERING. 


13 


Fundamental Torsion Formulas—Load 


When a bar is in torsion there is present an acting moment tend¬ 
ing to produce this torsion and a resisting moment which holds the 
acting moment in equilibrium. 

The acting moment is due to the force P acting at the end of the 
radius R, and is therefore expressed as P R. 

The resisting moment is expressed by the product of stress times 
polar moment of inertia divided by the distance from the neutral 
SJ 

axis, or-. 


c 

Then 


SJ 

PR = - 


And 


c 


SJ 

P — -, the fundamental formula for load. 

c R 


Fundamental Torsion Formulas—Deflection 

If a load, P, acts upon a common or simple torsion spring of 
length, l, to produce a deflection, then the angle of deflection, or ten¬ 
sion, may be represented by 0. If next we consider two sections of 
this bar Sx apart, then the angle of torsion, or relative angular dis¬ 
placement will be 

PR 

d — -5 x 

JG 

Then by integration the whole angle is 

PRl 

0 =- 


JG 


and, since P = 


SJ 


c R 

we have, by substitution, 


SI 


0 = 


c G 


Then since the deflection, /, equals the product of the radius times 

SRI 

the angle of deflection, or f — R © we have f — -, the fundamen- 


cG 


tal formula of deflection. 


Classification of Springs 

We are now ready to survey the entire field of springs, a dia¬ 
gram of which shows all the different forms as being related and 
built upon the fundamental forms. 










14 


SPRING ENGINEERING. 



The diagram herewith shows the field of commonly used springs. 
It can clearly be seen that more complex forms are but combinations 




SPRING ENGINEERING. 


15 


or variations of those in the diagram. As known commercially the 
more common forms are: 

1. Simple rectangular spring (constant section cantilever.) 

2. Simple triangular spring (constant strength cantilever). 

3. Leaf spring (combinations of the above). 

4. Ordinary spiral spring (clocks, etc.). 

5. Helical rectangular-bar torsion spring. 

6. Helical round-bar torsion spring. 

7. Simple rectangular-bar torsion spring. 

8. Simple round-bar torsion spring. 

9. Helical rectangular-bar compression spring. 

10. Helical round-bar compression spring (common). 

11. Conical rectangular-bar compression spring (volute). 

12. Conical round-bar compression springs (valve spring). 

13. Spiral spring, made of round bar (seldom used). 

The range of springs may be understood when one considers the 
heavy elliptics of railroad work weighing a quarter ton or more and 
the smallest of wire springs which require 38,000 to make one pound 
—19,000,000 to equal the weight of the larger spring. 



No. 1. (Uniform section cantilever.) 



No. 2. (Uniform strength and depth 
cantilever.) 


I- 

I_ 





-> 


No. 3. (Combination of above.) 







No. 4 or No. 13. (Rectangular or ellip¬ 
tical bars.) 


SPIRAL SPRING W 















16 


SPRING ENGINEERING. 





No. 5 or No. 6. (Rectangular or 
elliptical bars.) 


No. 7 or No. 8. (Rectangular or 
elliptical bars.) 


No. 9 or No. 10. (Rectangular or 
elliptical bars.) 


No. 11. (Volute.) 






















SPRING ENGINEERING. 


17 


CHAPTER II. 

ELLIPTIC SPRINGS 


It is doubtful if scientific calculations ever entered into the design 
of the original forms of such springs as are used under ordinary road 
carriages. Satisfactory as they are, they are not engineering results, 
but accepted standards born long ago of the cut-and-try methods of 
the blacksmith shop. Their manufacture belongs to such arts as are 
taught by father to son, or acquired through years of experience, dur¬ 
ing which have been gathered the “tricks of the trade.” The manu¬ 
facturer of this class of springs does not attempt to arrive at results 
by mathematics. He has learned as a part of his trade that certain 
styles of carriages should have certain springs. 

Sufficient time did not exist during the development of railroad 
cars for a gradual development of definite types of springs for various 
types of cars. It devolved, therefore, upon the engineer to design 
these springs; but as soon as the spring maker found that the 70,000,- 
80,000-, and 100,000-pound capacity car each had its own peculiar set 
of springs, and that any car could be fitted with springs according to 
its capacity, he adopted the engineer’s designs as another class of 
standards. Railroad cars, while resting on springs whose dimensions 
were originally scientifically estimated, are now, therefore, suspended 
largely upon springs belonging to a few fixed classes. 

With the advent of the automobile came a carriage traveling fast 
over uneven country roads, meeting severe usage in inexperienced 
hands, and demanding the extreme of comfort and safety. The ques¬ 
tion of springs and spring suspension thus becomes of primary im¬ 
portance, so that in these carriages each particular design requires a 
specially designed suspension. Automobile springs are fundamentally 
cantilevers, the same as all leaf springs. This class of springs more 
readily lends itself to an easy vibration, as well as to a better general 
design of the machine. It is possible to carry a load on a narrow- 
leafed elliptic leaf spring where there would not be room for a helical 
spring. Also, the addition of a leaf to an elliptic leaf spring adds to 
its capacity without changing its deflection, while the addition of a 
coil to a helical spring does not change its capacity but adds to its 
deflection. 

Any leaf spring, tightly banded around the middle, should be con¬ 
sidered as composed of two cantilevers of length l, where l is one-half 
the distance from center to center of the end bearings less one-half the 
width of the band. The length of each cantilever is then expressed 
(see Fig. 13) : 

c — w 


I — 


2 





18 


SPRING ENGINEERING. 


To consider a spring as a simple beam of length c, is to overlook 
the effect of the band. It is easily demonstrated that variations in 
the width of the band cause corresponding variations in the strength 
and deflection of the spring. The elliptic spring, graduated through¬ 
out, with but one leaf in each section extending from end bearing to 
end bearing, is fundamentally a cantilever of uniform strength; and 
the formulas applicable are based on the fundamental formulas of 
that type of cantilever. An elliptic spring with all leaves in each sec¬ 
tion extending from end bearing to end bearing is, on the other hand, 
a cantilever of uniform section, and the formulas for this type of 
cantilever are then applicable. 

The springs used in automobile practice are frequently combina¬ 
tions of these two forms, inasmuch as a considerable portion of the 
leaves extend the full length from bearing to bearing. It follows that 
neither of the above formulas will apply, but that the applicable form¬ 
ulas may be derived by combining the fundamental formulas for the 



Fig. 13. Diagrammatical Sketch of Graduated Spring, giving Length 
Notation used in Formulas 


two types of cantilevers. The load capacity of a cantilever is not 
affected by its form, for in either case: 

Sbh 2 

P = - 

61 

in which P — load, 

S = allowable stress, 
b = width of beam, 
h == thickness of beam, 
l = length of cantilever. 

In other words, the load capacity is equal for like conditions, such 
as stress, size of beam, and length of span. 

A great difference exists, however, in the deflections under the 
same load, one being fifty per cent more than the other: 

4 PV 

f — -, for uniform section cantilevers, 

Ebh a 
_ 6PT 

f — -, for uniform strength cantilevers,* 

Ebh 5 

in which f = deflection, and E = modulus of elasticity. 

* The formula given is that for a cantilever of uniform strength, where the height h is uni¬ 
form, but the width of the section of the cantilever decreases towards the outer end- b is the 
width at the support. 











SPRING ENGINEERING. 


19 


When such a difference as this exists, it is rather remarkable that 
many engineers calculate the properties of an elliptic spring no mat¬ 
ter what the cantilever conditions as though all elliptic springs were 
subject to the same rules and formulas; but, as a matter of fact, the 
proportion of back leaves, or the leaves on the longer side of the 
spring, which commonly extends the full length, ranges from 5 to 50 
per cent of the total number of leaves. It is not unusual to see at¬ 
tempts made through actual tests of the springs themselves to find 
the proper constant with which to modify the uniform strength equa¬ 
tions so as to render them applicable to springs composed of uniform 
section cantilevers in combination with uniform strength cantilevers. 
The desired modifier, however, is a variable quantity, depending upon 
the relative size of the fundamental spring elements. 

Lack of due consideration of this combination of different canti¬ 
levers accounts also for the different and conflicting formulas which 


p 'l 

- p'+p" 

i M 

p ,' 

= p 



I 



< ; : 



i 



N 


\ 

' 

1 1 


1 f - ' 



i I : 

i 

1 ! 

l 

i 

, i 

i i i 

I 



Fig. 14. Showing Division of Spring into Cantilevers of Uniform Section (Upper Portion) 
and Cantilevers of Uniform Strength (Lower Portion). One of the Full Length 
Leaves should always be considered as a Part of the Graduated Leaves 

various authorities advance. Thus Goodman, in “Mechanics Applied 
to Engineering”; Reuleaux, in his “Constructor”; and “Des Ingenieurs 
Taschenbuch” (Hiitte), give formulas all of which reduce to uniform 
strength cantilevers. Molesworth and the Automotor Pocket Book 
base their formulas on uniform section cantilevers. Henderson, who 
assumed all semi-elliptic springs to contain one-fourth full length 
leaves, and made an approximation of the result, was the first to rec¬ 
ognize the influence of the combination of cantilevers. 

Deduction of General Formulas 

For further consideration we will adopt the following notation, 
discussing only the semi-elliptic spring: 

P = total load on spring, 

Pi = portion of load on one end of spring, 

P' = portion of load on one end of full-length leaves, or on 
uniform section cantilever, 




















20 


SPRING ENGINEERING. 


P" — portion of load on one end of graduated leaves, or on uni¬ 
form strength cantilever, 
n = total number of leaves, 
n' = number of full-length leaves, 
n" — number of graduated leaves, 
n' 

r =—, 
n 

S — maximum fiber stress in spring, 

S' = maximum fiber stress in full-length leaves, 

S" = maximum fiber stress in graduated leaves, 

/ = total deflection of banded leaves, 

/' = total deflection of full-length leaves if unbanded, 

/" = total deflection of graduated leaves if unbanded, 
b = width of leaves, 
h — thickness of leaves, 
l = length of cantilever, 

L = net length of spring, i. e., actual distance between end bear¬ 
ings, less width of band, 

x =proper initial space between fundamental cantilevers before * 
banding. 

It is but reasonable to assume that the maximum fiber strain 
should be the same in both fundamental parts, or 


But 


S' = S". 

6 P’ l 
S' = -, 

n'bh 2 


Hence, 


6 P" l 

S" = - 

n" b h 2 

P' n' 


P" n " 

In a well-designed spring there should be, at full load, a division 
of the work proportional to the respective number of leaves in the two 
fundamental parts. The fundamental formulas of the two cantilevers 
have shown, however, that such proportional loads would produce dif¬ 
ferent deflections in their respective carriers. This difference in de¬ 
flection would cause a separation of the two portions of the spring 
were they initially together and unbanded. Were they initially to¬ 
gether and banded the result would be internal stress under load 
which would mean that a division of the load proportional to the re¬ 
spective number of leaves in the two fundamental parts could not 
exist. 

It is evident that by placing a space between the two fundamental 
parts when unloaded and unbanded, equal to the difference between 
the two deflections, there will result no space between the two funda¬ 
mental parts at full load; and hence if banded in this position there 






SPRING ENGINEERING. 


21 


will be no internal stress, so that the load on each part will be propor¬ 
tional to the number of leaves in that part. If then the load be re¬ 
moved, it follows that bhe band alone holds the two portions together 
and that there must exist a resulting stress upon the band and leaves. 
Now 

4 P'Z 1 2 3 

r= - (i) 

En'bh 3 

and 


6P" Z 3 

/" =- ( 2 ) 

E n " b h 3 

But, as shown, 

P' n' 


or 


Hence /' = 
Hence, 

Also, since 


P" n" * 
n' P" 


-, as derived by substituting in (1). 

E n" b h 3 

2 P" Z 3 

/" — /' = -. 

E n" b h 3 

P ' P" P, P 


we have 


Also since 


or 


2 n 


/"-/'= 


PZ 3 

Enbh 3 


L 


1 9 

2 

PL 3 

= -• 

% Enbh 3 

PL 3 


8 E nb h 3 


This last expression is then a general expression of the proper 
initial distance between the two fundamental portions before banding, 
expressed in terms of total load on spring, total number of leaves in 
spring, and net span of spring. To find the actual working deflection 
of the entire spring it is only necessary now to ascertain how much 
either portion is deflected by the process of bending. For this purpose 
let us adopt the following notation: 

Px = force exerted by band, 

fx = deflection of full-length leaves caused by band, 

/x" = deflection of graduated leaves caused by band. 














22 


SPRING ENGINEERING. 


Then, 

Hence 


or 


But 

Hence 


But 


Hence 


or 


or 


A' : 


2 Px r 


and /*" = 


3 Px l* 


En' bh 3 E n" b h 3 

P x l 3 V n' A" n" 

Ebh 3 2 3 

2 / 1 — r 




f " 
/* 


A' + A" : 


PZ 3 


Enbh 3 




PZ 3 


Enbh 3 


PI 3 


A" 


+ r / Enbh 3 
3 PxZ 3 


P w" 6 h 3 


3 Px Z 3 


E n" b h 
3 Px Z : 


= (—) 

V 24-r/ 


PZ 3 


+ t / Enbh 3 


Px = 


/ Sr \_ P 

V 2 -f r / P w 

f r(1 - r) A 

V 2 + r / 


P (1—r) wH 3 \2 + 
r (1 — r) 


6 /i 3 


The expression inside the bracket in the above equation becomes 
zero for either extreme value of r, as would be expected, the extreme 
values of r being unity and zero. The formula gives the force exerted 
by the band, i. e., the load upon the band. 

The total deflection of the graduated leaves, as already devel¬ 
oped, is, 

3 PZ 3 

r= - 

Enbh 3 


The deflection of the graduated leaves, caused by the band, is 


A" 



PZ 3 

Enbh 3 
























SPRING ENGINEERING. 


23 


The difference is, therefore, the deflection left in the graduated 
leaves after banding, or the general formula sought for the deflection 
of such a spring: 


or, 


/"-/*"= -J3 — 

/ 


L 

or, since l =— and 
2 


(i£)l 

-(^ 7 ) 


PI 3 


Enbh 3 


p = 2 Pi = 2 


+ r / Enbh 3 


/ Snbh 2 \ 

\ 61 ) 


Hence 


( 6 

/ 2 S n b h z \ 

1 L ’ 

V 2 + r / 

V 3 L ) 

1 8 Enbh 3 


2 (2+r) 


X 


SL 2 

£ 7/1 


This last expression is then a general formula for the deflection 
of all semi-elliptic springs. If all the leaves are graduated, r — 0, 
and 

SV 

/ — 1/4 X-• 

Eh 

If all the leaves are full length, r— 1, and 

SV 

/ — 1/6 X-• 

Eh 


As was to be expected, the spring composed of all graduated 
leaves has a deflection, according to the above general formula, 50 per 
cent above that of a spring composed of all full-length leaves. For 
values of r above zero, the deflection will be found to decrease until r 
equals unity. 

General Remarks 

The general formulas given above were first deduced by the 
writer in the early part of 1905, at which time they were placed be¬ 
fore Prof. C. H. Benjamin, then of the Case School of Applied Science, 
with a view of making extended experiments for the preparation of a 
thesis. It was the intention to have springs built with initial space 
as deduced, and compare the actual deflections of such springs with 
the estimated deflections. Although these experiments were not car¬ 
ried out, they are mentioned because it is believed that when such 
experiments are made, they will prove valuable. The deduction of 
the formulas was published for the first time in January, 1910. This 
deduction was made in connection with certain springs which were 















24 


SPRING ENGINEERING. 


giving very poor service, although designed by the same formulas as 
other elliptic springs. It was the writer’s conclusion that had the 
springs been built with the proper initial space between the funda¬ 
mental parts, these springs would not have broken, and that the omis¬ 
sion of this space caused an over-stress in the full-length leaves, and 
an under-stress in the graduated leaves, which caused the over¬ 
strained leaves to break, throwing an overload upon the previously 
under-stressed leaves, which also broke when the stress became exces¬ 
sive. This conclusion seems to explain why springs of this type are 
frequently found with only the long leaves broken; the remaining 
leaves, all being of one type, divide the resultant overload evenly so 
that the over-stress is not so excessive. Perhaps the strongest indi¬ 
cation of the correctness of the deduction lies in the well-known fact 
that the percentage of breakage is always much greater with semi- 
elliptic springs (of the combination type, usually) than with full 
elliptic springs. Also, it is generally found upon unbanding these 
springs that no initial space exists. 

Comparison of deflections estimated from the above formula, with 
actual deflections, has in some cases been quite satisfactory, while in 
other cases the actual deflections have appeared closer to those esti¬ 
mated by uniform strength formulas. In such cases where the writer 
has been able to make comparisons, however, the springs had been 
made to specified deflections which evidently ivere estimated by the 
uniform strength formulas. Experienced spring makers know that it 
is quite possible by putting a “pull” in the springs to vary the deflec¬ 
tion and load. This trade term, “pull,” is itself nothing more nor less 
than the introduction of an initial space between the leaves before 
banding. 

Calculations of Springs 

The calculation of spring properties by formulas is long and te¬ 
dious. The writer appends, therefore, a table based on a modulus 
of elasticity of 25,400,000 and a fiber stress under maximum safe load 
of 80,000 pounds per square inch. Calculations of springs made of 
materials having other physical properties are made by simple pro¬ 
portion. This table is to be used only when all leaves are fully grad¬ 
uated. 

The safe load on one leaf one inch wide is found by dividing the 
constant given under Pu by the net length. The corresponding de¬ 
flection is found by multiplying the constant given under /«., by the 
square of the net length. 

Example: What is the safe load on a semi-elliptic full gradu¬ 
ated spring of five leaves if of one-quarter by two inch steel; length 
between end bearings, thirty-six inches; band or seat, three inches? 

Net length = 36 — 3 = 33 inches. 

3333.33 

Load on one leaf one inch wide =F-= 101.01 pounds 

33 






SPRING ENGINEERING. 


25 


Semi-Elliptic Spring Table 


Giving safe load and deflection for 1 inch wide leaves, 1 inch net length. 
Used only when all leaves are fully graduated. 


Thick¬ 
ness of 

Pu 

A 

Steel 

Pu 

/« 

Leaf 





52.08 

0.02519 

T2 

4218.75 

0.00280 

* 

208.33 

0.01260 

T6 

5208.33 

0.00252 

■h 

468.75 

0.00840 


6302.08 

0.00229 

Vs 

833.33 

0.00630 

Vs 

7500.00 

0.00210 

T2 

1302.08 

0.00504 

1 3 

64 

8802.08 

0.00194 

~16 

1875.00 

0.00420 

7 

T6 

10208.33 

0.00180 

■h 

2552.08 

0.00360 

15. 

32 

11718.75 

0.00168 

Va 

3333.33 

0.00315 

y* 

13333.33 

0.00157 


Load on one leaf two inches wide = 2 X 101.01 = 202.02 pounds. 
Load on five two-inch leaves = 5 X 202.02 = 1010.10 pounds. 
Corresponding deflection is: 

0.00315 X (33) 2 = 3.43 inches. 

Formulas can easily be deduced making it possible to use the ac¬ 
companying table for other classes of elliptic springs than those of 
the semi-elliptic type with all leaves fully graduated. 

The formulas for the semi-elliptic spring with all leaves gradu¬ 
ated are: 

2 Snbh 2 SL 2 

P = - and / =-. 

3 L \Eh 

To find the values of Pu given in the table, insert S = 80,000, 
n= 1, 6 = 1, ft = the value given in the first column in the table, and 
L — 1. To find the values of /u, insert in the second formula S = 
80,000, L = l, E = 25,400,000, and h — the value given in the first 
column in the table. 

Now if the values in the table are to be used for other springs, 
constants can be deduced by which the table values may be multi¬ 
plied. 

For a semi-elliptic spring with a portion of the leaves graduated 
the load P remains the same as for a spring with all leaves graduated. 
The formula for the deflection, however, is: 

1 SV 

f = -X-. 

2(2-fr) Eh 

The values in the table, therefore, must be multiplied by the 
2 

quantity- X L 2 to find the deflection for any given combina- 

(2-f-r) 

tion full leaf and graduated spring of effective length L. 

For a full elliptic spring with all leaves graduated, P still re- 




















26 


SPRING ENGINEERING. 


mains the same as for a semi-elliptic spring, but / doubles its value, 
or: 

SL 2 

/ =- • 

2 Eh 

The values in the table, therefore, in this case must be multiplied 
by 2 L 2 . 

For the full elliptic spring with only part of the leaves graduated, 
the load P remains the same as before, but the deflection is twice that 
of a semi-elliptic spring: 

1 2 SL 2 SL 2 

/ =-X-=-. 

2 (2 + r) Eh (2 + r) Eh 

In this case, then, the values for the deflection in the table are to 
4 

be multiplied by- X L 2 . 

2 + r 

The flexibility of a spring is the amount of deflection as com¬ 
pared to the load. This may be expressed as so many inches deflection 
per hundred pounds, or y. 

Example: Assume a full-elliptic, fully graduated spring, where 
S = 80,000, 

E = 25,400,000, 
b = 1% inch, 
n = 4, 

h= 1 A inch, 

L = 30 inches. 


Then the safe load equals: 
P — 4X 1% X 


3333.33 


30 


778 pounds. 


And the deflection equals: 

/ = 30 2 X 2 X 0.00315 = 5.67 inches. 

Then, 

5.67 

y =-X 100 = 0.73 inch. 

778 


On the other hand, assume that the thickness and number of 
leaves are unknown. Then we have: 

P = 778 pounds, 

5 = 80,000 

E = 25,400,000, 

6 = 1% inch, 

L = 30 inches, 
y = 0.73 inch. 

778 

Then, f = -X 0.73 = 5.67 inches. 

100 









SPRING ENGINEERING. 


27 


But / — 2 /»L 2 , where /u is the constant for deflection in the ac¬ 
companying table. 

Hence, 

/ 5.67 

/u =-=-= 0.00315 

2 L 2 1800 

The thickness of steel in the table which corresponds to this 
value of fu is one-fourth inch. 

The number of leaves is found by using P». 

Load on one leaf, one inch wide is: 

3333.333 

-= 111.11 pounds. 

30 

Load on one leaf 1% inch wide is: 

111.11 X 1% = 194.25. 

Number of leaves is then, 

778 

- =z4. 

194.25 

The present calculation makes no allowance for the leaves of a 
spring varying in thickness. Where such springs are used, the deflec¬ 
tion of the different leaves will not be uniform. Hence, in such springs 
also a suitable initial “pull” should exist, and such springs should be 
estimated by a general formula based upon a combination of different 
cantilevers, thus making allowance for different depths of cantilevers. 
It is much better to use springs composed of but one thickness of 
leaves, as the combination of different thicknesses adds a complexity 
scarcely necessary. 

Results obtained from fully graduated full elliptic springs would 
seem to show that the action of the friction between the leaves is not 
great enough to seriously affect the bending action, in that the formu¬ 
las give results agreeing very closely with actual conditions. 

It should be noted that the above discussion does not take into 
account the effect of nesting leaves one inside the other. As a matter 
of strict fact the leaves of a leaf spring, having thickness, are not free 
to assume the same radius of curvature, i. e., the actual radii must of 
necessity vary from one leaf to the next by the thickness of the leaf 
itself. The consideration of this element adds greatly to the complex¬ 
ity of the problem, and comparisons of actual tests with results de¬ 
duced by the above formulas seems to indicate that on heavy springs 
at least such consideration would be an unnecessary refinement. 





28 


SPRING ENGINEERING. 


CHAPTER III. 


SPIRAL SPRINGS 


Under spiral springs come all flat spirals, such as clock springs, 
watch springs, etc., also those springs called helical torsion springs. 
The latter are unlike the former in that the radius of the bar wind¬ 
ing is constant instead of increasing. This constant radius makes it 
necessary for the bar to assume the helical form. The action upon 
the bar in both cases, however, is one of bending instead of torsion. 

The formulas generally given are those of Realeaux, but it should 
be noted that these formulas are based on the supposition that l is 
the length of the spring bar and that R represents both the ra¬ 
dius of application of the load and the radius of curvature. That such 
is the assumption is shown in the derivation of these formulas which 
follows. In order to arrive at the first expression for deflection we 
have the following: 

P is the load. 

D 

-is the leverage or radius. 

2 


(4-H 2 7r = linear movement 

X4X 2 7rwj= Energy expended 


or deflection, 
by bending mo¬ 


ment in twisting spring through n turns. 


But P 



And M — 


El 

R 


Also 2 ir n = 9, the total angular movement 

'(-fX-M^XO 


f = OR = 

tion. 


PDR(2vn) PDl Ml 

e = -=- =- 

2 EI 2EI El 

MIR PR 2 l 

-=-, the general expression for deflec- 

EI El 


Here l is not, as assumed, the length of the bar which forms the 
spring, for tt D n is simply the linear expression for the deflection, 
which becomes equal to the length of the spring only if the spring 
becomes a straight bar when released. In other words if a tempered 
straight bar were bent so that the force applied followed the bar and 
actually moved through the linear distance, v D n, then tt D n would 
become the expression for length of bar as well as deflection of load. 










SPRING ENGINEERING. 


29 


Also, if in one case R is the radius of the bending moment, and 
in another the radius of curvature, then this assumption means also 
that the spring is a straight untempered bar when released of all 
load; otherwise R cannot be the radius of curvature. 

The formulas of Reuleaux, the derivation of which follows, are 
therefore not applicable to a spring which is of spiral or helical 
shape when not under stress. The attempts to fit these formulas to 
such springs has led to much confusion. The writer knows of no 
formulas which have yet been developed which are intended to apply 
to unstressed spiral or helical shapes used under torsional loads. The 
derivation of such formulas and comparison of same with practical 
results should be a field for interesting research. 

The general expression for load is the simple fundamental form¬ 
ula 

SI 

M =-. 

c 

From the above we proceed as follows: 

Circular Bars—Deflection of Spiral Springs 
PR 2 1 

As before, f — - 

El 

7 rd A 64 PR 2 1 

But 1 = - / =- 

64 t rd*E 


Circular Bars—Load of Spiral Springs 

SI SI SI 

As before, M — - And PR = - And P — - 

c c c R 

d 7 r d* 7r d 3 S 

But, c = — And, 1 = - P = - 

2 64 32 R 


Rectangular Bars—Deflection of Spiral Springs 
PR 2 1 

As above, / —- 

El 


b h 3 

But, 1 = - 

12 


12 PR 2 1 
Ebh s 


Rectangular Bars—Load of Spiral 

SI SI 

As above, M — - PR - 


But, c — — 
2 


And, I 


bh 3 


12 


Springs 

SI 

And P = - 

cR 

Sbh 2 

.*. P = - 

6R 




















30 


SPRING ENGINEERING. 


CHAPTER IV. 

HEAVY HELICAL SPRINGS 

A spring is usually specified by three dimensions, although some 
specifications complete the design by a fourth. The dimensions usually 
given are the outside diameter, free height, and diameter of bar. The 
fourth dimension, the solid height, is not generally given, so that the 
actual design of the spring is really left to the manufacturer. In 
some cases the number of coils or “rings” is specified, but this should 
never be done, as a tapered coil may be considered by one as a full 
coil and by another as a partial coil, thus causing confusion. 

Investigation of such formulas as are found in the general text¬ 
books, hand-books, and books of reference, indicates the need of more 
direct formulas to facilitate the design of springs. It is the writer’s 
intention to present the derivation of such formulas with parallel ex¬ 
amples, showing the ease of application. For this purpose we adopt 
the following notation: 

d — diameter of bar, 

D = mean diameter of coil, 

/ = total deflection, 
h = solid height, 

H = free height, 

L = blunt length of bar, 

W — weight of bar, or spring, 

P = capacity of coil, 

Pi = any load less than capacity, 
hi = height of coil under load Pi, 

S = maximum fiber stress, 

G = torsional modulus, 
w = weight of steel per cubic inch. 

Only round bar coils will be considered. 

I. Length of Bar When Solid Height is Given 

L 

Total number of coils =-. 

V D 
h 

Total number of coils =-. 

d 

Hence, L h 

Example: Outside diameter = 4% inches, 

Bar = 7/16 inch, 

Solid height = 10 inches. 

/m \ 

L = 3.1416 X ^) X 10 = 282.74 inches. 






SPRING ENGINEERING. 


31 


II. Deflection When Solid Height is Given 

] mdamentally, as given in most text-books, 



Example: Outside diameter = 4% inches, 

Diameter of bar = % inch, 

Solid height = 10 inches. 

( 3V 2 \ 2 

-I X 10 = 4.34 inches. 

% / 

III. Ratio Between Free and Solid Heights 


Hence, 


H — h 


H 


Or, for steel springs, 


H = h + f 

■-' S | 


G ' 

l d , 

i " s 1 


G * 

K d 

^r S i 
+- I 

f D ' 


H = I 1 + 0.019946 


(-,)] 


and 


H 


1 + 0.019946 


( 1 ) 


Example 1: Outside diameter = 6 inches. 

Diameter of bar = 1% inch, 

Free height — 13% inches. 

Find solid height h. 

13.75 

h =-= 10 inches. 


1 + 0.019946 


(-) 

\ 1 % / 










32 


SPRING ENGINEERING. 


Example 2: Outside diameter = 7 Vs inches, 
Diameter of bar = 1 Vs inch, 
Solid height = 10 inches. 

Find free height H. 


H= £ 1 + 0.019946 (^ 7 -) 


X 10 = 15.67 inches. 


IV. Deflection When Only Free Height is Given 

ir S / D 


f 


(v)* 


But 


U 


Hence, 


Or, for steel springs 


IT S / 
1 + - ( 

g \ 

< D \ 2 

< d ) 

G / D ' 

y- 

TT S \ d y 

7T S / 

1 + - ( 

G \ 

D \ 2 

d ) 

H 


G ( 

1 +- ( 

ITS V 

D \ 2 

d ) 

>» 

H 


1 + 50.1337 

(+' 


Example: Outside diameter = 5V 2 inches, 

Diameter of bar = 1% inch, 

Free height = 11% inches. 

11 % 

/ —-*— = 1 % very nearly. 


1 + 50 


/i% y 

.1337 (-1 

\4 Ys / 









SPRING ENGINEERING. 


33 


V. Weight When Solid Height is Given 


7r d 2 


Area of cross section = 


4 


Lit d 2 


Cubical contents of bar = 


4 


Lit d 2 V) 


Then W = 


4 



dDh 


Hence, W = 


4 


For steel springs, where one cubic foot of steel weighs 486.6 


pounds, 


. W = 0.694 dDh. 

Example: Outside diameter = 3% inches, 


Diameter of bar = 15/16 inch, 

Solid height = 10 inches. 

15 13 

W = 0.694 X — X 2 - X 10 = 18.3 pounds. 

16 16 

VI. When Free and Solid Heights are Given to 
Determine Stress 


H 


h — 




Example: Outside diameter = 4% inches, 


Diameter of bar = V 2 inch, 


Free height = 22% inches, 


Solid height = 10 inches. 


12.75 / 0.5 \ 2 


S = 4,010,700 X- (- I — 80,000 pounds. 

10 V 4 / 


5 = 4,010,700 X 













34 


SPRING ENGINEERING. 


VII. When Free and Solid Heights are Given 
to Determine Capacity 

STTef 

P = - 

8 D 


and 


Hence, 

For steel springs, 



Gfd* 

P —- 

8hD\ 


fd 5 

P = 1,575,000- 

hD 3 


Example: Outside diameter = 2% inches, 

Diameter of bar = % inch, 

Free height = 14 Vz inches, 

Solid height = 10 inches. 

4.5 X 0.5 5 

P= 1,575,000 X-— 1653 pounds. 

10 X 2.375 s 

These last two formulas are very useful in ascertaining the 
stresses and loads of the separate coils of double and triple coil 
springs. 


VIII. Given Free Height, Diameter of Spring and Bar, and 
Load Carried at Given Height. To Find 
Proper Solid Height 

Pi fi 


P f 
H = f + h 
H = f 1 + h 1 

Hence, fi = f + h — h t 
Then P ( f+h — ht) =P>f 

Pj — Pf+Ph* 

Hence h = - 

P 


Pi — P 

h = -X / + h t 

P 


But 



h 


TT S / 


Hence, 


D 


Pi — P 










SPRING ENGINEERING. 


35 


1 + 

For steel springs, 
h — _ 


irS / P — P 1 


(^m 


1 + 0.019946 


C^)(4) 


Example: Outside diameter = 5% inches, 

Diameter of bar = % inch, 

Free height = 18 inches. 

What solid height is required for carrying 1395 pounds at 14 
inches? 

Sn d? 

P = 2970 pounds by formula P = 


Then, 


8 D 


14 


1 + 0.019946 


( 2790—1395\ /4% V 
2790 / V % / 


10 inches. 


IX. To Determine the Quality of the Steel 

The value of G is the index to the quality of the steel, and upon 
this value depend all properties of the spring. By transposing the 
formulas given in (VII) for stress and load we find a method for as¬ 
certaining this value, i. e.: 



Example: Outside diameter = 4% inches, 

Diameter of bar = 11/16 inch, 

Load = 1219 pounds, 

Deflection = 3.7 inches, 

Solid height = 10 inches. 

10 X (4&) 8 

G = 8 X 1219 X-= 12,600,000. 

3.7 x (ny 


General Remarks 

Concentric coils, as shown in Fig. 18, are made generally of the 
same free and solid heights. Presuming that such coils are all made 

D 

of the same quality of steel, the ratio of — should be the same 

d 

throughout, for the formula in (II) clearly shows that this is neces¬ 
sary to obtain equal stresses in all coils. 

D 

The formula in (1) shows that when all values of — are made the 

d 

same, the lengths of all bars will be the same before tapering. A 














36 


SPRING ENGINEERING. 


D 

study of all the formulas reveals the fact that the ratio of — deter- 

d 

mines everything; the writer emphasizes the importance and meaning 
of this ratio by terming it the spring index. 

The absolutely perfect design of concentric springs is seldom pos¬ 
sible where a scale of sixteenths inch for dimensions is used, with the 
customary one-eighth inch between inside diameter of one spring and 
outside diameter of the next. As cases of perfect design, however, 
the following springs are given as examples: 

Outer: 5 inches outside diameter, 15/16 inch bar, index 
Inner: 3 inches outside diameter, 9/16 inch bar, index 4J/*. 



Fig. 15. Typical Heavy Helical Springs 


Outer: 2% inches outside diameter, % inch bar, index 6. 

Inner: 1% inch outside diameter, V* inch bar, index 6. 

In concentric coil springs where perfect design is impossible, the 

D 

coil having the least value of — will be stressed the highest, as shown 

d 

by the formula in (VI) ; this coil may therefore be called the govern¬ 
ing coil, inasmuch as the motion, or deflection, of the spring as a 
whole depends upon this coil. To estimate the capacity of such con¬ 
centric coils we have recourse to the formula in (VII), while the for¬ 
mula in (VI) shows the separate stresses. The load which the concen¬ 
tric spring will carry at any height is then found by the fact that all 
loads are proportional to deflection. 

In actual design adjacent coils are wound in opposite directions to 
prevent binding, as shown in Fig. 18. Instead of using concentric 
coils, groups of similar coils are sometimes used which are held to¬ 
gether by pressed steel or cast spring-plates, as shown in Fig. 16. It 
is customary to suspend the static load at one-half the deflection. 















SPRING ENGINEERING. 


37 


CHAPTER V. 

GROUPED HELICAL SPRINGS 

It is the intention to present here, a study of the design of 
grouped helical springs, developing the subject upon the basis of 
the relation which exists between the diameter of the bar and the 
mean diameter of the spring. In the discussion only round bar coils 
will be considered. 

Notation 

The following notation will be adopted: 

S = stress solid, or maximum stress, usually assumed to be 
80,000 pounds per square inch for heavy steel springs; 

G = modulus of torsional elasticity, taken at 12,600,000 pounds 
per square inch for steel springs; 



Fig. 16. Groups of Coil Springs held together by Plates at Top and Bottom 

w — weight of one cubic inch of the spring material; 

7T = 3.1416; 

/ = total deflection; 

H = free height; 
h — solid height 
/ii= any other height; 

p = capacity at solid height, or weight necessary to produce com¬ 
plete deflection; 

Pjizrload at hi, or weight necessary to compress to 
W = weight of spring; 

L = blunt length of bar, or length before tapering; 
p = mean diameter of coil; 
d = diameter of round bar; 
r = spring index, or D H- d. 




38 SPRING ENGINEERING. 


Definitions 

In addition to the above notation, the following definitions will 
serve to clear the discussion: 

Spring: any single coil, combination, or group of coils. 

Coil: a spring composed of one bar only. 

Turn: a wind or rotation, a part of a coil. 

Turns are fundamental elements; coils are composed of winds; 
and springs consist of one or more coils. 


The Spring Index 

The deflection of a helical spring may be expressed as 



The capacity may be expressed by 



The weight may be expressed as 


v2 w D 2 h 

w — - 

4 



( 1 ) 

( 2 ) 

(3) 


The length of bar to form the spring may be expressed 


L — tt 



h 


(4) 


These four standard equations being solved, the length of bar 
required to make the spring will be known, as well as the spring 
weight, capacity and deflection. A further inspection of these formu¬ 
las will show that all properties depend upon the ratio between the 
diameter of the bar and the mean diameter of the spring. This all- 
important ratio we have, in the preceding chapter, termed the spring 
index, expressed as 

D 

d 


Fundamental Principle of Grouped Springs 


Equation (2) gives one value of —, while equation (3) affords 

D 

another. Equating, 

8 P 4 W 


Whence 


7 r S d? n 2 w D 2 h 


WS 

P =- 

2 7T w h 



(5) 


(6) 









SPRING ENGINEERING. 


39 


This is the fundamental principle of grouped spring design and 
means that when a constant weight of material is uniformly stressed, 
the resultant capacity varies inversely as the square of the spring 
index, and that the actual number of coils or dimensions thereof is 
immaterial for a constant weight and spring index. 


To Ascertain the Value of the Spring Index 


Having given the desired capacity, free height, solid height and 
material of a spring, it may further be assumed that the maximum 
fiber stress and modulus of elasticity are also known. If then, the 
spring index be ascertained, the ratio of mean diameter of coil to 
diameter of bar that must be maintained in order to produce the 
results desired, will thus be given. 



Fig. 17. Double and Triple Coil Concentric Springs 


The value of the spring index from equation (1) is, 



(7) 


which may readily be solved since f — H — h , the difference between 
two known quantities. 

Constant Areas, the Basis of Bar Sizes and Dimensions 

No matter of how many bars or coils the spring unit may be com¬ 
posed, the sum of the cross-sectional areas of the individual bars is 
constant. This fact furnishes a basis from which to ascertain the 
sizes and dimensions of the bars, according to whether there is one or 




40 


SPRING ENGINEERING. 


more coils used. It is important, therefore, to deduce an expression 
for this constant area. Consider a single coil spring. 


The product of equations (1) and (2) is 

t r 2 S 2 dDh 

Pf = - 

8 G 


which may be expressed 

S 2 / t r 2 dD h \ 
Pf ~ 2G ( 4 ) 

(8) 

Then 

t r 2 dDh 2 GPf 

(9) 


Equation (3) may now be written 

/ 7 t 2 dDh \ 

W = i --- \ w (10) 

TT'dDh 

Substituting the value of-as given in equation (9) 

4 

2 GPfw 

W = - (11) 

S 2 


The total weight divided by the unit weight will give the volume, 


or 

W 2 G P f 

V = -=-- 

w S 2 

From equation (4) the length of the bar will always be 

L — irrh (12) 

The volume, divided by this constant length, will therefore result 
in an expression of the constant area, or 


V _ _ 2 GPf 

L t rS i rh 


(13) 


Substituting from equation (1) the value of /, placing 


gives 


2 Pr 

A= - 

S 


D 

d 


— r > 


(14) 


which is the constant area value sought, and being in known terms 
may be readily obtained. 

This applies equally well for a multi-coil spring, for the weight is 
uniformly taken up by each of the units of the spring. Therefore 
the total cross-sectional area is constant. 

















SPRING ENGINEERING. 


41 


Determinate Equations for Bar Sizes and Dimensions 


In concentric coils, let all the properties of the inner coil be de¬ 
noted by the subscript 1; of the next coil by the subscript 2; of the 
next coil by the subscript 3; and so forth. The total sectional area 
will then always be 



irdn 2 2 Pr 

— = — (15) 

4 S 


which may be expressed 
7r 


— (dx’+ek’-Ms 2 -!- 


2 Pr 

+ da 2 ) —- 

S 


(16) 


It is possible also from the relation of the diameters of the coils 
to form as many equations as there are coils less one, so that there 



Fig. 18. Double and Triple Coil Concentric Groups, showing Right-hand 
and Left-hand Coiling, to prevent Binding 


may be always found as many equations as there are unknown quanti¬ 
ties, or bars, or values of d. 

Equations based on the relations of coils are deduced as follows, 
where Dn ==' inside diameter and Dn " = outside diameter of nth coil. 


Da' -j- da - dn ^^ -j- da 

(17) 

Da" = (r+ 1) dn 

(18) 

Dn — (r - 1) dn 

(19) 


or 

In the same way 


Then let the difference between the outside diameter of one coil 
and the inside diameter of the next be taken as any desirable clear¬ 
ance, c. Or 


Dn' — D"u~ 1 = C 


( 20 ) 














42 


SPRING ENGINEERING. 


This gives the series of equations sought, thus: 

Between first and second coils, 

(r—1) <k= (r + 1) di-\- c (21) 

Between second and third coils, 

(r—1) d 3 = (r + 1) do -f- c (22) 

Between third and fourth coils, 

(r —1) d .= (r+ 1) d 3 + c (23) 

and so forth. 



Fig. 19. A Concentric Group showing what is meant by “Solid,” “Loaded” and “Free 
Heights.” The Clearance between Coils is usually one-sixteenth inch 


Equation of d x for Single Coil Spring 

The value of di f or the diameter of the inner (in this case the 
only) bar may be obtained by writing equation (16) simply as: 

7T dx* 2 P 

-1=- r (24) 

4 S 

which may be readily solved for di after making the proper numerical 
substitutions of the other quantities, thus: 

8 P 

d? = - 

T S 


( 25 ) 






































SPRING ENGINEERING. 


43 


Equation of for Double Coil Spring 

If there are reasons for desiring two coils in the spring, equation 
(21) gives 



and from equation (16) 

7T 2 P 

— (d 1 2 + d i 2 )= - r (27) 

4 S 

Substituting the above value of d 2 in this gives an equation which 
after substitution of constants may be readily solved for d h after 
which di and the outside diameters may be readily found. This sub¬ 
stitution results in 



Equation of dj for Triple Coil Spring 

If now three coils are desired, from equation (21) as before: 
r + 1 c 

cU =- di H- 


and from equation (22) 


r— 1 


r + 1 

d, — - di 


whence 


r — 1 


r — 1 
c 

r — 1 


di 


(^) 


r -f 1 c 

di +- c -|- 


(r-1) 


r — 1 


Then from equation (16) 

— (dx 2 + di 2 + di-) 

4 

whence 

r + 1 c 

cM 


SG 


(29) 

(30) 

(31) 

(32) 


( r - f-1 c \ 2 /(r + 1) 2 r+l c \ 3 

-dx +-) + (-dx +-c +-) :} 

r — 1 r — 1/ V (r — l) 2 (r — l) 2 r — 1/ 

8 P 
- r 


irS 


(33) 


Equation of for any Number of Coils 


From equation (14) it is apparent that if there be n number of 
coils, the nth. value of d , or d n , is always 

r -f- 1 c 

da —-- djx—i -|- ( 34 ) 


r — 1 r — 1 

The general formula for di may therefore always be written, 
although every additional coil adds greatly to the complexity of the 
final expression. 

























44 


SPRING ENGINEERING. 


Obstacles in the Design of Concentric Coils 

The increasing complexity of the equation of di, offers an obstacle 
to the solution of the multi-coiled springs on strictly mathematical 
lines. A still greater obstacle in the use of the formulas deduced 
lies in the fact that commercially it is found economically practical to 
use only such sizes of bars as are commonly rolled by the mills. In 
the absence of tables giving the properties of various spring coils 
from which a selection may be readily made, it is believed that the 
above formulas, (25), (28), (33) and other similar formulas which 
may be readily deduced, will serve as guides to the best commercial 
sizes to use, which sizes being once determined may then be investi¬ 
gated and their future combined action ascertained with certainty. 
To make the manner of proceeding clearer, assume a definite problem. 


Solution of Problem by Foregoing Formulas 

Problem: To ascertain the proper coils to use to support 35,464 
pounds at 5.022 inches solid height, the free height to be 6.625 inches. 
From equation (7) the spring index is 


I (6.625 — 5.022) 12,600,000 

r=\ - 

v 3.1416 X 80,000 X 5.022 

whence r = 4, closely. 

Size of Bar for One Coil Spring: 

By equation (25) 

8 X 35464 X 4 

dS = - 

3.1416 X 80,000 

whence di~2y 8 inches. 

Therefore D =4d t = 8^ inches 
and D" = 5d x = 10 % inches. 


Size of Bar for Two Coil Springs: 

Assume the usual clearance of 1/16 inch between coils, whence 
c=y 8 inch. From equation (28) 


or 


d> 2 +( JU + -) 

V 3 24/ 


2 _ .8 X 35464 X 4 
3.1416 X 80,000 


34 10 1 

— diM- di-\ -= 4.5156 

9 72 576 


whence di = 1.07 inch and d, = 1.825 inch. Therefore DP = 5.35 
inches, and DP = 9.125 inches. 








SPRING ENGINEERING . 


45 


Result of Adopting Bars of Commercial Sizes 


The closest commercial sizes to the above decimal solutions would 
then be 


inch, and d 2 — 111 inch, and 
DP = 5% inches and DP = 9 Vs inches 
The actual value of c is then (see notation on page 41). 

D 2 ' — DP = 5V 2 — 5% = Vs inch 

Now turn the investigation to the two coils which have been se¬ 
lected. The spring index of the inner coil will be 


Of the outer coil, 


Dx 4t 5 s 69 

r, =-=-= - = 4-1/17 

dx 1* 17 

D 2 7& 117 


r 2 — -=-=-= 4-1/29 

d 2 111 29 


It may now be seen that although the design was based on a 
constant spring index, the limitations of practice and economy have 
rendered it impossible to maintain this ideal condition. As the spring 
indexes of the inner and outer coils are of different values, it is known 
at once that the deflections and lengths of bar will not be identical for 
the same solid height. This means that commencing with the same 
free height and compressing to the same height will cause one coil 
(that having the least value of spring index) to be stressed higher 
than the other. 

If the value of the spring index has been diminished only slightly 
from that assumed, it is a safe assumption that the fiber stress will 
be increased but slightly beyond that assumed, in which case it is not 
necessary to calculate the actual stresses, but the real capacities may 
be arrived at directly by basing the calculations upon the modulus of 
elasticity. It is more satisfactory, however, to ascertain the fiber 
stresses also, and where the value of the spring index has been con¬ 
siderably altered such a course is imperative in order to keep within 
safe limits of stress. 


Solution of Actual Problem—Stresses 

The results will be the same whether similar free heights be 
taken and compressed to the same maximum fiber stress, or whether 
a beginning be made with the same solid height extending to the 
same maximum stress; the maintenance of a uniform final stress 
results in final heights which are not uniform. Instead of different 
final heights the usual practice is to use uniform free and solid 
heights, with the result that each coil is then stressed differently as 
pointed out before. 

In this case the actual stress in each coil is found by the formula 

Gf / d \ 2 






46 


SPRING ENGINEERING. 


which is simply an expression of the fact that where the material 
used, and the free and solid heights are uniform, the stress varies 
inversely as the square of the spring index. 

In the particular problem at hand, the stress in the inner coil 
would then be 

6.625 — 5.022 


51 = 4,010,695 
and in the outer 

5 2 = 4,010,695 
Or, since 


5.022 

6.625 — 5.022 
5.022 


then 


Sz 


O 

> - <:) (:.) 

C) • 

( 29 \ 2 / 69 \ 2 

m) = 


700 pounds 


78,600 pounds 


600 pounds 


which is the same as before. 

The stresses being known, the load on each coil may now be 
solved by the following formula: 

TrSd 3 


P = 


8 D 


Solution of Actual Problem—Capacities 

When the deviation from the assumed index is slight, the varia¬ 
tion in the maximum stress will be correspondingly small, and the 
experienced designer is therefore safe in proceeding to estimate the 
capacity of his spring directly from dimensions and without reference 
to actual stresses. In this case use the formula 

Gfd 6 

P = - 

, ^ . 8 li D 3 

or, where G is 12,600,000 for steel springs 

/ <f 

P = 1,575,000- 

This would give for the inner coil 


Pi = 1,575,000 
and for the outer, 

P 2 = 1,575,000 
The capacity of the two 


6.625 — 5.022 (1.0625) 5 

5.022 (4.3125) 3 

6.625 — 5.022 (1.8125) 6 

5.022 (7.3125) 8 

coils together will then be 


Pi-h Pa — P — 33,640 pounds 


= 8490 

= 25150 













SPRING ENGINEERING. 


47 


Some idea of the difference which exists in the theoretical and 
practical design may now be gained from Table I, which makes a 
detailed comparison. 


Table I. Comparison Between Estimated and Actual Coil Spring Results 



Estimated 

Actual 

Free height 1 

6.625 

5.022 

1.603 

80,000 

80,000 

35,464 

1.07 

1.825 

5.35 

9.125 

6.625 

5.022 

1.603 

77,700 

78,600 

33,640 

1.0625 

1.8125 

5.375 

9.125 

Solid height ^ Assumed same. 

Deflection J as actual . 

Stress, inner coil. 

Stress, outer coil. 

Capacity. 

Diameter inner bar. 

Diameter outer bar. 

Outside diameter inner coil. 

Outside diameter outer coil. 


Limitations of Concentric Grouping 

It is now apparent that in a spring concentrically arranged the 
inner bars are properly the smaller, and the greatest load is naturally 
upon the outer. There is a point, however, beyond which more inner 
coils will cease to be of advantage owing to the small gain in capacity. 
The addition of outer coils is also soon limited by the impossibility of 
coiling and tempering large bars. It is therefore evident that the 
load which may be carried by the concentric group is limited. 

Spring Plate Groups 

Where greater capacity is desirable than can be obtained by 
concentric grouping, several single coils, or several concentric groups, 
may be held together between spring plates of malleable cast iron or 
pressed steel. Such groups naturally offer greater stability than con¬ 
centric groups; but, where the concentric group affords sufficient ca¬ 
pacity and stability it should be used, as it is more economical of space 
and does not necessitate the use of spring plates to hold the different 
coils together. As the load should be supported firmly upon the cen¬ 
ter of the unit, the group should be arranged with such symmetry that 
the supporting forces, or spring resistances, will balance about any 
axis. 

The designing of groups of this kind consists in the simple oper¬ 
ation of dividing the load into as many parts as there will be units 
in the group. Then, maintaining the desired free heights and solid 
heights, and hence the same constant spring index, proceed to design 
the separate units in the manner just presented for the simple con¬ 
centric. Ordinarily much time and labor may be saved by remember¬ 
ing that halving the diameters of bar and coil reduces the capacity 
and weight to one-quarter, but does not affect length of bar or de¬ 
flection of coil. This is due to the fact that this really halves the 
spring index with effect as indicated in formulas (1), (2), (3) and 
(4) page 38. 




























TABLE II. COMPARISON OF FOUR COIL SPRING GROUPS FOR SAME CAPACITY 


48 


SPRING ENGINEERING 



4^> CO O 

o o £ 

*-« a £ 

bJO a) a 

£ * £ 


£ 

(D 

0) 

to 


_rH ^ 

ID CO 


£ ^ 
0) o 

O) *rH 

£ -e 

£ 


CD 

2 X 

£ -m 


4-5 

£ 




CO 

• M 


0) 

£ 

O ' 
co r! 

•c ~ 

§.■3 
S -o 
° J 2 

o ^ 
£ 


a> 


CO 

CO 

o 

ft 


a> 

r£ 


CO 


I 

CO 

o 

ft 

CO 

0) 

r* 

£ 

s 

r£ 

o 

• rH 

X 

£ 

o 

co 

13 

co 

0) 

CJ 

£ 

"O 

<D 

*N 

CO 

£ 

rO 

0) 

X 

-4-5 


^H 

£ & 
^ 3 

a £ 
H 

£ co 
*43 bo 
£ C 

£ 

M ^ 
p CO 

73 *£ 
a) 


• 4^> 
CO o 

■+5 £ 

£ Q. 

s I 

9 s 

bJQ 

£ 


O 

<D 

N 

• 

w bo 
£ 


<D 

r£ 

Eh 


ft 
. £ 

i 2 5 


<3 > 

• rH 
£ 

cr* 

0) 


4-5 CS 

£ rrH 

£ 2 
> p 

• rH 

£ a> 

C 1 *-* 
0 ) £ 


a> 

4-> 

4-> 


£ 

£ a> 
£ 

£ co 




















































































SPRING ENGINEERING. 


49 


CHAPTER VI. 

CONICAL HELICAL SPRINGS 


On account of their physical characteristics, conicaJl helical 
springs divide themselves into two distinct classes, according to their 
use. The formulas applicable to a spring of this type used as a com¬ 
pression spring are not at all applicable if the spring is to be used 
as an extension spring. This is more obvious if we note that the 
safe load which an extension spring will carry is governed by the 
capacity of the largest or weakest coil, which condition is reversed in 
the compression state, where the spring retains its flexibility until the 
load becomes great enough to close up the smallest or strongest coil. 
It is the object of this article to develop the formulas applicable to 
the various types of conical helical springs. Round bar coils only 
will be considered. 

Notation—Dimensions in Inches, Weights in Pounds 

S — stress, 

G = modulus of torsional elasticity, 

/ = deflection, 

/<f= deflection under load Px 
H = free height, 

h = solid height, assumed to equal d X IV, 
y = solid height at any convenient coil, 

P = capacity of spring, 

Px — any load not exceeding P, 

Pi = capacity of largest coil, 

P 2 —capacity of smallest coil, 

A = mean diameter of largest coil, 

Di — mean diameter of smallest coil, 
d — diameter of bar from which coil is made, 

A = mean radius of largest coil, 

B — apex height of cone, 

C — mean radius of smallest coil, 

N — number of coils, 
p — average horizontal pitch of coils, 
w = weight of one cubic inch of steel, 
l = length of bar in spring, 

W ~ weight of spring, 
x = mean radius of any convenient coil. 

Deflection and Capacity of Extension Spring 

It is obvious that Px in this case must be understood to be not 
greater than the capacity of the weakest coil, inasmuch as a greater 
value of Px would distort the spring beyond the realm of rational 
formulas. A conical spring is composed of an infinite number of 
elementary cylindrical coils, each element being in itself a uniform 
diameter helical spring, and each element differing from its neighbor 
in the one respect that each successive element has a mean diameter 




50 


SPRING ENGINEERING. 


infinitesimally less than its first neighbor, and likewise infinitesimally 
greater than its neighbor on the other side. These increments of 
change in the mean diameter result in corresponding increments of 
change in the deflection of the successive elements, and being all gov¬ 
erned by the one general expression for deflection of cylindrical heli¬ 
cal springs, may be added together by resorting to calculus. Since 




Fig. 21 


Diagrams for the Derivation of Spring Formulas 


the increments of change in the mean diameter are in this case in 
proportion to the increments of change in the solid height, it follows 
that the increments of change in the deflection also follow those of the 
solid height, and that we may expect to arrive at the summation of the 
deflection through a summation of the increments of change in the 
varying solid height, which for successive elementary cylindrical coils 
increases from 0 to its maximum h. 

Now the expression for deflection in cylindrical coil springs is: 


and for capacity: 



TT S d 3 

P = - 

8 D 


The value of D is here, however, the variable, represented by 2x. 
Therefore, in each elementary cylinder: 

/* P* 8Px(2x) 3 h 64 Px x 3 h 

— — —y or /* =-=- 

f P Gd 5 G cf 

Further, 


64 Px 

8 /x —- x 3 By 

G (f 


f 1 

; 


i>64 Px 64 Px 

- x 3 B y — - 

Gd? Gd? 


f. x ' Si j- 


































SPRING ENGINEERING. 


51 


B A Ay 

But, as shown in Fig. 20,— =-; whence x — A - 


Hence x 3 = A 3 ( 1 
Hence, 


( 


A — x 
3 y 3 y 2 


-) 

B 3 / 


U 


64 Px A* r* h / 

Gd* J 0 \ 


B B 2 


3 y 3 y 2 y 3 


B a 

64 Px A 3 / 3 h 2 h 3 

fx — - y 1 -—+ 


Gd 5 


■)' 

j.) 


2 B B 2 4 B 3 
A h 

Also, from Fig. 20, we have: B — -, whence: 


64 Px A J 
G(f 


( 


A —C 

3 h 2 (A — C) h 3 (A — C) : 
/?,-1- 


(A — cy 


2 Ah 


A 2 h 2 


4 A 3 h 3 


D 1 D, 

But A —- and C~- -. Hence, 

2 2 

8 Px D 3 h / 3 (D 1 — D 2 ) 


Gd 5 


( 


(D 1 — D 2 ) 2 (D t — D 2 y 


2 Z)i 


Bx 2 


4 Z>i 3 


) 


8 Px Px 3 /Px 3 + Px 2 P 2 + Px P 2 2 + P 2 3 


( 


4 Px 3 


) 


Gd 5 

2Pxh 2 Pxh (Px 4 —P 2 4 ) 

- (Px 3 + Px 2 P 2 + P, P 2 2 + ^ 3 ) =- 

Gd 6 (Px —P 2 ) 


Gd 8 


Px (Px 4 —P 2 4 ) 

(Px —P 2 ) d 

Gd 1 - 

2h 

Observe now that the expression in the denominator is the aver¬ 
age horizontal pitch p (see Fig. 20) of the coils: 

Px — P 2 


2 h (Px — P 2 ) d 

p =-; but iV =—; hence, p — - 

N d 2 h 

Therefore, 

^ _ Px (Pi 4 - Do*) 

fx Gpd* 

This, then, is the formula for the deflection of any extension type 
conical helical spring under load Px, the value Px not exceeding the 
capacity of the weakest coil, equivalent to the capacity of a cylindrical 
helical spring of a mean diameter equal to Px and bar diameter equal 
to d. 




























52 SPRING ENGINEERING. 


If the value of Px equals the capacity of P of this spring, we have: 

ir Sd* 

P — p x =- 

8 Di 

Substituting this value of Px, we have the total deflection: 

ttS {D/ — DS) 
f =- 

8 G p dDt 

which is the final formula for the total deflection of the extension 
conical helical spring. 

It will be noticed that the previous discussion is based on the 
assumption that the solid height of the spring is equal to the diam¬ 
eter of the bar times the number of coils in the conical spring. How¬ 
ever, as one coil seats within the other, the solid height is really less 
than that assumed; the solid height becomes less as the taper of the 
coil becomes greater, until for a true spiral spring the solid height is 
reduced to the diameter of the bar. The actual deflection is dependent 
on the “slant height” of the conical coil which is equal to d X N, 
as assumed for the solid height, rather than upon the actual vertical 
height. Due to this assumption, the changing of the angle of the cone 
formed by the spring does not change the actual deflection so long as 
the value of the slant height itself is not changed. Inasmuch as the 
above discussion is based on the slant height, the actual solid height 
of the spring and also the free height of the spring should be correct¬ 
ed by deducting from the value of these heights the difference between 
the slant height and the solid height. Stated briefly, the assumption 
that the solid height is equal to d X N is necessary in order to obtain 
the correct value of the deflection; but after this deflection has been 
obtained, correction should be made for this assumption. 


Deflection and Capacity of Compression Spring 


The deflection of a compression spring of this type is a funda¬ 
mental problem of the same type. In this case, however, the summa¬ 
tion is not the summation of increments of deflection under a uniform 
load and varying stresses, but the summation of increments of deflec¬ 
tion when each elementary spring is stressed to a maximum, and hence 
under uniform stress. 


In this case 


TT S 

Sf = - 

G 


__ ttS / D\ 2 7 rS / 2x\ 

f -I I h, becomes / = -1- 1 h 

G \d ) G\d) 

( 2 x \ 2 /^i. irS /2x \ 2 

_) 5 „and/=/ o — (—)«» 


But x — A 


A y 


B 

4 7r S A 


Hence: 


G 


S A p / 

d 2 J 0 \ 


4 7 TS A 2 
Gd 2 


( 1 

2 y 

+ — 

V 

B 

B 2 , 

( h 

h 2 

h a \ 

V 

B 

SB 2 / 


) 













SPRING ENGINEERING. 


53 


Ah D x D 2 

But B =-, and A =-, and C — -. Hence the expression 

A —C 2 2 

above can be transformed to: 

ir S h 

f =-(Pi 2 + Di D, + P 2 2 ) 

3 Gd 2 

_ 7T S' (D 1 3 — P 2 3 ) (Pi — Zb) d 

/-, and since p =-- 

3 Gd 2 (D 1 — D,) 2 h 

_ t rSiDS — D./) 

6 G p d 

which expresses the total compression for a conical helical compression 
spring. 

The capacity, solid, equals that of the smallest coil, or: 

7 T ScP 

p =- 

8 Dz 

Deflection of Compression Spring for Given Loads 

If the given load is less than Pi, the entire spring remains flex¬ 
ible, and the formula for deflection is the same as that derived for an 
extension spring, the condition of varying stress still being present. 

If, however, the load exceeds Pi, then a portion of the spring 
will become solid. The division point may be found, for Px — 
7 rSd? 

-, and since the load in cylindrical coils varies inversely as D : 

8 Dx 

D x P, D,P, 

-=-, whence Dx — - 

Di P X P X 

The deflection of the two portions of the spring should now be 
considered separately and added, using the two final formulas just 
developed. 

Fig. 21 shows a graphical illustration of the divided spring. 
Let p x be the average horizontal pitch of the unclosed portion, above 
Dx. Let p y be the average horizontal pitch of the solid portion, below 
Dx. 

The total deflection is then: 

7T S (Pi 3 — Px 3 ) 7 fS (Px 4 — P 2 “) 

/ —- 1 - 

6 G d py 8 G d Dxpx 

In Fig. 21, hy is the height, solid, of the solid portion of the spring. 
Its value is derived thus: 

Hy Pi-Px Pi -Px 

-=-, whence h y = h -- 


h 


Pi — P,. 


Pi — P 2 

















54 


SPRING ENGINEERING. 


Fundamentally 
spring 


Bar Length for Conical Spring 

h. so that we have in a conical 


■Ml) 


2 7T 

l = - I x s II— 


l 

2 A V r u / y\ 2 A 7T / h 2 \ 


5 y, or 


Dr Ah D 2 

And since A =-and 5=-, and C —-, we get: 

2 A—C 2 


7 r h / Dr + \ 

=— (— ) 


Weight of Conical Spring 

l TT c? W 

Fundamentally, W =-. Hence 

4 

ir 2 dhw 

W = - (Dr + D 2 ) 

8 


Summary of Formulas 

Summarizing and substituting for S a value of 80,000 pounds 
per square inch, and for G a value of 12,600,000 we have: 

Dr — D* 

Deflection; / = 0.002493-for extension spring 

p d Dr 

Dr 3 — D 2 8 

/ = 0.003324-- for compression spring 

p d 

d 3 

Capacity; P — 31,416-for extension spring 

Dr 

d 3 

P — 31,416-for compression spring 

D 2 

Weight; W = 0.35 dh (Dr + D 2 ) in each case 
h 

Bar length; Z = 1.571— (Dr + D 2 ) in each case 
d 















SPRING ENGINEERING. 


55 


Numerical Examples 

Example: Compression spring, Zb = 4 9/16 inches; D 2 = 3 9/16 
inches; d = 1 1/16 inch; h — I 1/16 inches. 

h Di — D 2 Di — D 2 

Then 2V = —=6.65; -= 0.5; p — -= 0.075. 


/ = 0.003324 


(«) ; 


2 

(3&) 5 


2 N 


= 2 iV approximately. 


0.075 X liV 

See the accompanying table of cubes, and also other powers of 
numbers required in conical spring calculations. 

H = 7 1/16 + 2 1/16 = 9 1/8 inches. 

Example: Same spring in extension. 

(4ft) 4 — (3ft) 4 

/ = 0.002493-= 1% approximately. 

0.075 X lft X 4ft 

H (extended height) = 7ft + 1% = 811 inches. 

As might have been expected, the free height for the compression 
type is greater than the possible extended length for the extension 
type. This is because sufficient load to fully stress the smaller or 
stronger coils cannot be applied without distorting the extension 
spring, whereas the coils may all be stressed to maximum stress in 
the compression type, the closing of the coils solidly together protect¬ 
ing the spring from over-stress. 


Reversion to Cylindrical Helical Springs 

It will be noted that in each of our final formulas we have in- 

Di — D 2 

troduced the factor ( Di — D 2 ) and the value of p 


-(—)*■ 
V 2 h / 


This has been done to leave the formulas in as simple a form as 
possible. Note, however, that if Di and D 2 each be taken as equal 
to Di or to each other, the formulas in each case revert to the funda¬ 
mental cylindrical helical spring formulas, as should be expected. 

Substituting Di — D 2 — D in the extension formula 

2 P*h 

f* =-( Di 3 + Di 2 D s + Di D 2 2 + DS) , we have 


/* = • 


G<f 
8 Px D 3 h 


G ef 

This is the fundamental formula for the deflection of an extension 
helical cylindrical spring under any load Px, derived directly from 
the formula for load: 

t rSd 3 

P = - 

8 D 











56 


SPRING ENGINEERING. 


in which S has been replaced by its value in the formula 


7T S 



Gfd " 


/ = 


G \d 


h giving P =- 

8 hD 3 


8 P D 3 


from which / —- h 


G<f 


Substitute, again, Di = D 2 = D in the compression formula: 


t rSH 


f =- {Di + Di D 2 -f ZV), and we have 

3 GcF 



the fundamental formula for the deflection of a compression helical 
cylindrical spring. 

The reason for this comparison between the formulas for cylin¬ 
drical helical springs and conical springs, is particularly, to bring 
out the fact that, while each of the conical formulas are thus shown 
to revert to the same form when Di = D 2 , yet the conical spring form¬ 
ulas themselves, for extension and compression springs, are different. 
As already mentioned, one is an expression for deflection under a 
given load, regardless of stress, while the other is the expression for 
deflection under a uniform maximum stress. The former condition is 
that of the conical extension spring, while the latter is that of the 
compression type. 


Auxiliary Formulas 


The expressions for deflection and capacity are the main form¬ 
ulas for all helical springs; from these are developed such other form¬ 
ulas as may be desired. In this particular case, care must be taken 
in making such further developments, if using the expressions ZV — 
ZV and ZV — D 2 , to note that resulting formulas will not revert to 
simple cylindrical helical formulas, because of the fact that a zero 
quantity has been introduced when Di becomes equal to D 2 . 

Therefore, further formulas are based on the longer but primary 
formulas which we have arrived at before the introduction of the 
quantity Di — D 2 . 








SPRING ENGINEERING. 


57 


The Ratio Between Free and Solid Heights 

Since H — /-}- h, we have, for compression springs: 
tt S h 

H = -(ZV+Z), D, + D■?) +h 


3 Gd 2 


= h 1 + 


7 rS /Z)r + D,D 2 -fZ)/ 




3 d 2 


)] 


[ 7 rS / D? — D? \1 

1+ G \3 <f (A — £>,) / J 


In a similar way, for extension springs: 

2 Px / Di + Pi 2 D, + Pt P 2 2 + P 2 3 


H = h |1 




G 

[ 2 Px / P, 4 — P 2 4 \ I 

G Vd 5 (Px — P 2 ) / J 


d 5 


)] 


= fc|l + 

G \d 6 (P 1 — P 2 ) 

By introducing the factor (Pi — P 2 ) the formulas are thus sim¬ 
plified as before, so that they may be readily solved with a table of 
cubes and fourth powers. 


Deflection When Only Free Height is Given 

Considering first the compression type, we substitute the value of 
h as found from the formulas in the last paragraph in the general 
formula for /. Hence: 


H 

H 

( ™ ^ H 

G /3 d 2 (Pi — Do) \ 

V Px 2 + Pi D, + P 2 2 / 

t rS \ Px 3 — P 2 3 / 


In a similar way, for extension coils: 

H 

f = - 


1 


G / d 5 

2 Px \ P/ + Pi 2 P 2 + Pi P 2 2 + P 2 3 


H 


G /d 5 (Px —P 2 )\ 
2T\ Di* — P 2 4 / 






















58 


SPRING ENGINEERING , 


General Considerations 


In the compression type it is sometimes desirable to fix the 
heights, both free and solid, and afterwards ascertain the resulting 
capacity. If the heights so fixed exceed the allowable deflection by the 
compression formula, the spring will not return to its original free 
height. In other words, it will have taken a set. If the difference 
in heights is less than that of the compression formula, it cannot be 
assumed that there will be a uniform stress throughout the spring 
when solid, as there would have been had the spring been built to the 
highest free height possible, and the capacity will not then be in pro¬ 
portion to the deflection. If the deflection, for instance, is one-half 
of the formula deflection, the capacity will not necessarily be one- 
half that of the strongest coil, instead of equal to that of the strongest 
coil. This type then appears indeterminable for capacity, the diffi¬ 
culty being to so pitch the coils as to assure uniform stress when the 
spring is solid. This difficulty does not present itself in cylindrical 
coils as we have a uniform stress at solid height. 

Uniform stress at solid height in a conical spring requires a pitch 
of coils in proportion to the deflection of same at maximum stress, or, 
which is the same thing, in proportion to the diameters of the various 
elements. As the diametrical increase per unit of bar length is not 
a straight line formula, the pitch of coils necessary to gain uniform 
stress when solid would have to follow the law of a definite curve. 
While it may be possible to develop a machine which will so pitch 
these elementary coils, yet the demand does not seem to have devel¬ 
oped such a machine. 

Where the deflection is made originally greater than the max¬ 
imum stress will allow, the first compressions of the hardened spring 
will reduce the deflection to the maximum which the steel will stand. 
Thus, in such a case there is an assurance of uniform stress. 


The laws governing the action of grouped cylindrical helical 
springs apply likewise to grouped conical springs. Briefly, the de¬ 
sign should maintain the same free and solid heights throughout, 


D x 

which means that for all coils in the group the — ratio should be 

d 


D 2 

the same, and the — ratio should likewise be the same for all coils. 
d 




SPRING ENGINEERING. 


59 


CHAPTER VII. 

WIRE SPRINGS 

Wire is classified by gage, a certain gage meaning a wire that 
will pass through a standard sized hole or slot. These holes or slots 
are numbered; thus a No. 10 wire will pass through the No. 10 hole 
in the gage. From this it will be clear that wires are not identified 
by their actual diameters as in the case of round bar stock. If there 
were only one wire gage in general use, this method of specifying 
wire sizes would be less confusing; but unfortunately there are sev¬ 
eral gage systems which were originally developed by workers in 
different trades or shops, and each of these is in more or less general 
use. The more common of these wire gages are: (1) American, or 
Brown & Shape; (2) Birmingham, or Stubb’s iron wire; (3) Wash¬ 
burn & Moen; (4) Stubb’s steel wire; (5) American screw gage; and 
(6) steel music wire. As a complete series of wire sizes are made in 
accordance with each of these systems, and as only the more widely 
used systems are mentioned in the preceding list, it will be evident 
that the different sizes of wire are numerous and that these sizes over¬ 
lap each other. 

Impracticability of a Complete Table 

In arranging a table of springs which are made from rods, the 
table may be laid out according to the sizes of the rods, i. e., 1/4, 
5/16, 3/8 inch in diameter, etc. Then the properties of the springs 
may be tabulated under each size of rod, according to the outside di¬ 
ameter of the spring, i. e., 4 inches, 4 1/16 inches, 4 1/8 inches, etc. 
In such cases intervals of 1/16 inch in the outside diameter of the 
spring or in the diameter of the bar from which the spring is made, 
is the smallest variation required for all ordinary cases. No such 
method can be followed in working out a table of wire springs, how¬ 
ever, because the different wire gages in common use are too numer¬ 
ous and the variations in the diameters of the springs are infinite, 
the spaces being very erratic and often consisting of infinitesimally 
small differences. Moreover, the variation of any dimension by even 
1/32 inch may be entirely too great when compared with the size of 
the spring, as some wire springs are so small that it requires several 
thousand of them to weigh one ounce. The best that can be done is 
to resort to a makeshift and develop an approximate table which will 
give a good idea of the properties of any given spring. In addition 
to the trouble resulting from different wire gages in common use, a 
further difficulty is encountered owing to the fact that springs are 
made of all kinds of material—such as aluminum, brass, bronze, steel, 
etc.—and there are no standard grades of wire for this purpose, as 
in the case of rods where the Pennsylvania Railroad specifications for 




60 


SPRING ENGINEERING. 


spring steel are recognized as a standard. On this account we can 
only base the data in a wire spring table on a certain specified fiber 
stress and leave it to the engineer to use that as a basis where a dif¬ 
ferent fiber stress must be used. 


The Spring Index 

No matter what the size of the wire or the diameter of the spring 
may be, it is always possible to arrive at the value of the ratio which 
we have termed the “spring index.” This ratio is given by the follow- 

“ g: D 


Spring index =— 
d 


where D = mean diameter of spring; 
d = diameter of wire. 

D 

A wire spring table containing all values of the spring index — 

d 

would, of course, be infinitely long and include infinitely small differ¬ 
ences, so that any practical table must omit many possible values. 
Consequently, reference to the table will often result in failure to 

D 

find the exact value of the spring index — that is wanted. In such 

d 

cases the nearest value will serve to give a fair idea of the actual 
spring properties, and will generally be close enough to answer all 
practical purposes. In order to use the spring table which is pre- 

D 

sented in this connection, the first step is to calculate the value of — 

d 

for the required spring, and also the value of d 2 . The second and 
fourth columns of the table contain constants which are multiplied 
directly by the solid height of the spring, but the values in the third 
and fifth columns of the table must be multiplied by d 2 . If the weight 
of the spring is required, it is also necessary to multiply by the solid 
height of the spring. The results presented in the table are based 
upon a value of 80,000 pounds per square inch for the fiber stress and 
a torsional modulus of 12,600,000. 

To make the method of procedure quite clear, we will calculate 
the properties of a wire spring to be made of spring steel which has 
a safe fiber stress of 80,000 pounds per square inch, the diameter d 
of the wire being 1/16 inch and the mean diameter D of the spring, 
3/16 inch. The solid height of the spring is to be 1 % inch. It is 
required to know: (1) the length of wire necessary to make the 
spring; (2) the weight of the wire in the spring; (3) the free 
height of the spring; and (4) capacity of spring. With this, we find, 

D 3/16 

Spring index — =-= 3 

d 1/16 
1 

d 2 = [1/16] 2 =-. 

256 








SPRING ENGINEERING. 


61 


Now referring to the table (Page 63) and a value of 3 for the 
D 


spring index, —, we find the required length of wire to make the 
d 


spring to be (from the second column of the table) : 

1% X 9.4248 = 11.78 inches. 

From the third column, the weight of this wire is found to be: 

-X 2.0973 X 1% =0.0102 pound. 

256 

From the fourth column, the free height of the spring is found 
to be: 


1% X 1.1795 = 1.47 inch. 

From the fifth column, the capacity of the spring is found to be: 
1 


Capacity -X 10,472 = 41 pounds per coil. 

256 


Initial Tension 


Springs are often made with an “initial tension” which causes the 
coils to be drawn tightly together. This result is secured by twisting 
the wire, a common example of a spring of this type being the or¬ 
dinary screen door spring. Such springs will not begin to deflect as 
soon as the load is applied, it being necessary to first overcome the 
initial tension already in the spring. With springs of this type it is 
possible to load to the maximum capacity without obtaining a corre¬ 
sponding deflection of the spring. 


Methods Used in Spring Manufacture 

Wire springs are made of a great variety of materials, and the 
wire is generally sold under a trade name. The manufacturer is 
very willing to guarantee his wire of certain grades, but equally un¬ 
willing to guarantee other grades of wire which he makes. In all 
cases, however, he either cannot or will not state any of the physical 
properties of his product, contenting himself by saying it is “extreme¬ 
ly uniform.” Some manufacturers actually disclaim the possession 
of such knowledge, while others are frank in stating that exact anal¬ 
yses and corresponding characteristics are regarded as valuable trade 
secrets. Some manufacturers of both wire and wire springs, even go 
so far as to state that they make certain grades of wire for their 
own use and that they will not sell such wire to other spring manu¬ 
facturers. In such cases it is necessary for the spring manufacturer 
and his engineer to ascertain by experiment just what any grade of 
wire will do. In this way an exact average of the physical properties 
of the wire is obtained. 

Wire springs are likely to be very small; for instance, the writer 
has one particular spring in mind which was so small that 38,000 
were required to weigh one pound. For measuring wire of any kind, 
it is better to use a micrometer caliper, thus recording the dimen¬ 
sions in decimal parts of an inch, instead of attempting to use gages 
of any kind. 




62 


SPRING ENGINEERING. 


Spring Ends 

Extension springs are furnished with an infinite number of differ¬ 
ent kinds of ends, many of which have become so commonly known 
in the trade that they are regarded as standard. The two hooks at 
opposite ends of a spring may be made in line and in the same or dif¬ 
ferent planes; and they may be located at the center of the spring 
or on one side. A further variation may be obtained by arranging 
the ends out of line and they may even be located at right angles to 
each other. Some common forms of spring ends are illustrated in 
Fig. 22, and in referring to these illustrations it will be well to 
remember that the direction in which the springs should be wound 
ought always to be specified when giving an order. It must also be 
borne in mind that tables and calculations of the form presented in 
this article apply only to the flexible part of the spring. Ends of 
various kinds, such as hooks, plugs, etc., are really not part of the 
spring proper, such ends being merely mechanical appliances for at¬ 
taching the spring to other machine elements. 



Fig. 22. The Spring A has a Regular Machine Hook over Cen¬ 
ter; B, Regular Hand Loop over Center; C, Double-coil Hand Loop 
over Center; D, Regular Hand Loop at Side; E, Small Eye at one 
Side; F, Small Eye over Center; G, Small Hook at one Side; H, Plain 
End; I, Ground End; J, Long Hook; K, Long Square Hook; L, V- 
hook; M, Loop knotted or secured to Spring; N, Square Loop knotted 
or secured to Spring; O, Knotted Eye; P, Extended Eye; Q, Straight 
End (usually annealed so that it can be twisted); R, Annealed End 
eyed and twisted; S, Tapered End with Extended Swivel Eye* T 
Tapered End with Regular Swivel Eye; U, Tapered End with Swivel 
Hook; V, Tapered End with Swivel Bolt; W, Plain End with Plus: to 
screw into Place; and X, Plain End with Hooked Plug. 
























































SPRING ENGINEERING. 


63 


WIRE SPRING TABLE 


Weight per inch of solid height equals A X d 2 . Capacity of coil equals B X d 2 , where 

d is the diameter of bar in inches. 


D 

d' 

Length per Inch 
of Solid Height. 

Weight per Inch 
of Solid Height. 
A. 

Free Height per 
Inch of Solid 
Height. 

Capacity. 

B. 

3 

9.4248 

2.0973 

1.1795 

10,472 

3* 

9.6212 

2.1410 

1.1871 

10,258 

3k 

9.8175 

2.1847 

1.1948 

10,053 


10.0138 

2.2284 

1.2027 

9,856 

3* 

10.2102 

2.2721 

1.2107 

9,666 

3f% 

10.4066 

2.3158 

1.2189 

9,484 

3§ 

10.6029 

2.3595 

1.2272 

9,308 

3j% 

10.7992 

2.4031 

1.2357 

9,139 

3£ 

10.9956 

2.4468 

1.2443 

8,976 

3^j 

11.1920 

2.4905 

1.2531 

8,819 

3! 

11.3883 

2.5342 

1.2621 

8,666 

3H 

11.5846 

2.5779 

1.2712 

8,520 

3f 

11.7810 

2.6216 

1.2805 

8,378 

O 3 

Oy 0 - 

11.9774 

2.6653 

1.2899 

8,240 

3| 

12.1737 

2.7090 

1.2995 

8,107 

Oyg; 

12.3700 

2.7527 

1.3092 

7,979 

4 

12.5664 

2.7964 

1.3191 

7,854 

4 * 

12.7628 

2.8401 

1.3292 

7,733 

4g 

12 9591 

2.8838 

1.3394 

7,616 


13.1554 

2.9275 

1.3498 

7,502 

41 

13.3518 

2.9712 

1.3603 

7,392 

4i% 

13.5482 

3.0149 

1.3709 

7,285 

4f 

13.7445 

3.0586 

1.3818 

7,181 

4% 

13.9408 

3.1022 

1.3928 

7,080 

4* 

14.1372 

3.1459 

1.4039 

6,981 

4 A 

14.3336 

3.1896 

1.4152 

6,886 

4$ 

1U.5299 

3.2333 

1.4267 

6,793 

4U 

14.7262 

3.2770 

1.4383 

6,702 

4f 

14.9226 

3.3207 

1.4500 

6,614 

4R 

15.1190 

3.3644 

1.4620 

6,528 

4} 

15.3153 

3.4081 

1.4740 

6,444 

4{i 

15.5116 

3.4518 

1.4863 

6,363 

5 

15.7080 

3.4955 

1.4987 

6,283 

5 r V 

15.9044 

3.5392 

1.5112 

6,206 

5* 

16.1007 

3.5829 

1.5239 

6,130 

5A 

16.2970 

3.6266 

1.5367 

6,056 

10 

5i 

16.4934 

3.6703 

1.5498 

5,984 


16.6898 

3.7140 

1.5629 

5,916 

v 10 

5f 

16.8861 

3.7576 

1.5763 

5,845 


17.0824 

3.8013 

1.5897 

5,748 











64 


SPRING ENGINEERING. 


WIRE SPRING TABLE 


D 
d * 

Length per Inch 
of Solid Height. 

Weight per Inch 
of Solid Height. 
A. 

Free Height per 
Inch of Solid 
Height. 

Capacity. 

B. 

5} 

17.2788 

3.8450 

1.6034 

5712 

5A 

17.4752 

3.8887 

1.6171 

5648 

5f 

17.6715 

3.9324 

1.6311 

5585 

5H 

17.8678 

3.9761 

1.6452 

5524 

5i 

18.0642 

4.0198 

1.6595 

5464 


18.2606 

4.0635 

1.6739 

5405 


18.4569 

4.1072 

1.6884 

5347 


18.6532 

4.1509 

1.7032 

5291 

6 

18.8496 

4.1946 

1.7187 

5236 

6^ 

19.0460 

4.2383 

1.7331 

5182 

6 

19.2423 

4.2820 

1.7483 

5129 


19.4386 

4.3257 

1.7636 

5077 

61 

19.6350 

4.3694 

1.7791 

5027 

6 A 

19.8314 

4.4131 

1.7948 

4977 

6| 

20.0277 

4.4567 

1.8106 

4928 


20.2240 

4.5004 

1.8266 

4880 

61 

20.4204 

4.5441 

1.8427 

4833 

6i% 

20.6168 

4.5878 

1.8590 

4787 

6f 

20.8131 

4.6315 

1.8754 

4742 

6ft 

21.0094 

4.6752 

1.8920 

4698 

6f 

21.2058 

4.7189 

1.9088 

4654 

6ft 

21.4022 

4.7626 

1.9257 

4612 

61 

21.5985 

4.8063 

1.9428 

4570 

6ft 

21.7948 

4.8500 

1.9600 

4528 

7 

21.9912 

4.8937 

1.9774 

4488 


22.1876 

4.9374 

1.9949 

4448 

71 

22.3839 

4.9811 

2.0126 

4409 

7ft 

22.5802 

5.0248 

2.0304 

4371 

71 

22.7766 

5.0685 

2.0484 

4333 

7ft 

22.9730 

5.1122 

2.0666 

4296 

7f 

23.1693 

5.1558 

2.0849 

4260 

7ft 

23.3656 

5.1995 

2.1033 

4224 

71 

23.5620 

5.2432 

2.1220 

4189 

7ft 

23.7584 

5.2869 

2.1407 

4154 

71 

23.9547 

5.3306 

2.1597 

4120 

7ft 

24.1510 

5.3743 

2.1788 

4087 

71 

24.3474 

5.4180 

2.1980 

4054 

7ft 

24.5438 

5.46i7 

2.2174 

4021 

71 

24.7401 

5.5054 

2.2370 

3938 

7ft 

24.9364 

5.5491 

2.2567 

3958 

8 

25.1328 

5.5928 

2.2765 

3927 

8A 

25.3292 

5.6365 

2.2966 

3897 

81 

25.5255 

5.6802 

2.3167 

3867 














SPRING ENGINEERING. 


65 


WIRE SPRING TABLE 


D 

d 

Length per Inch 
of Solid Height. 

Weight per Inch 
of Solid Height. 

A 

Free Height per 
Inch of Solid 
Height. 

Capacity. 

B 


25.7218 

5.7239 

2.3371 

3837 

81 

25.9182 

5.7676 

2.3576 

3808 

8 

26.1146 

5.8112 

2.3782 

3779 

8f 

26.3109 

5.8549 

2.3990 

3751 

8175 

26.5072 

5.8986 

2.4200 

3723 

81 

26.7036 

5.9423 

2.4411 

3696 

8 A 

26.9000 

5.9860 

2.4624 

3669 

8| 

27.0963 

6.0297 

2.4838 

3642 

8 rl 

27.2926 

6.0734 

2.5054 

3616 

81 

27.4890 

6.1171 

2.5271 

3590 

8tf 

27.6854 

6.1608 

2.5490 

3565 

81 

27.8817 

6.2045 

2.5711 

3540 

8H 

28.0780 

6.2482 

2.5933 

3515 

9 

28.2744 

6.2919 

2.6156 

3491 



Fig. 23. Group of Wire Springs 











































66 


SPRING ENGINEERING, 


CHAPTER VIII. 

HELICAL SPRINGS OF CONSTANT DIAMETERS 
BUT VARIABLE ELLIPTICAL AND 
RECTANGULAR BARS 


We wish in this article to investigate the effect which the va¬ 
riation of one dimension of the bar has upon the spring properties. 
The outside and inside diameters of the spring will first be kept con¬ 
stant, which will result in keeping D constant and also that dimension 
of the bar upon which it is not wound, the side of the bar which we 
see as we look at the end of the spring; while the side of the bar upon 
which it is wound will be varied. We will express the unchanging 
O.D.— I.D. 

bar dimension, or the - by d and the variation of the 

2 

other side we will express in terms of d as x d, x being a variable. 

It will then be clear that when x equals unity we are dealing 
with sections such as circles, squares, etc.; when x exceeds unity we 
are winding the bar upon the flat; when x is less than unity we are 
winding the bar upon its edge. Thus: 



Fig. 24. 


Springs made of flat material are useful where either a great 
amount of compression is required or where the load springs are to 
carry at some compressed length is such that round or square ma¬ 
terial cannot be used in their manufacture. 

Springs made of flat material coiled on edge are used where 
springs of considerable strength are required, and where springs are 
required to compress into a short space. 

Springs made of round and square material are sometimes 
coiled with all the pitch or space between coils which they will stand, 
and when in use they are often worked very rapidly to almost their 
closed length. Such springs are not durable but it is possible to con¬ 
struct springs which have the same strength but are made of flat 
material coiled on edge. Because of the extra number of coils which 
can be put into this type of spring, the strain on each coil is not as 














SPRING ENGINEERING. 


67 


great as it would be had round or square material been used, and a 
much more durable spring is the result. 

Springs are often required to carry a given load at some speci¬ 
fied point of compression, and yet the space left for the spring, i. e., 
the inside and outside diameters, is such that in order to give the 
strength required, the springs cannot be made from round or square 
material. In such cases, springs made of flat material coiled on the 
flat side are used. 


Elliptical Section—Load 

The polar moment of an elliptical section whose outside dimen¬ 
sions are a and b is for ordinary sections approximately: 

/ (bcf+ab 3 ) \ 

J = T \ 04 / 

From which the polar moment of an elliptical section whose out¬ 
side dimensions are d and x d is, 

7T d 1 

J = - (x -f a? 3 ) 

64 

Also in an elliptical section the distance from the neutral axis 
to the remotest fiber will be 
d 

c — -— when x equals unity or less than unity. 

2 

And 

x d 

c — — when * is more than unity. 

2 

The twisting moment to which any bar is subject is 

SJ 

T — - 

c 

We then have two expressions to derive for this twisting mo¬ 
ment, as follows: 


1. For x equal unity or less, we have 

T=w (7)(tS> + -' ) 

irSd* 

=- (x + X s ) 

32 

But also we know that, 

T = P R or in the case of a spring of this kind 

PD 

T=- 

2 

PD t rSd? 

-=-(a? -f as 8 ) 

2 32 


ttS d 3 

P = - (x + a: 3 ) — 

16 D 












68 


SPRING ENGINEERING 


Which may be expressed as 

IT S OT 

P = -Xx — 

16 D 

where Xi = x -f- 

Or for steel, 

d 3 

P=Yi — 

D 
7 rS 

where Yi = - Xi —15708 Xi 

16 

2. In like manner if x is more than unity, 




tf 3 ) 


it Sd 3 


32 


( 1+0 


PD 


7T s d 3 

p =— (1 + 0 — 

16 D 

Which may be expressed as 

ir S <P 
P = -Xx- 


In which case 
Or for steel, 

7T S 


16 D 
X,= (1 + x 3 ) 
d 3 

P=Yi 


D 


where Y x = -Xx ^ 15708 Xi, as before. 

16 

Elliptical Section—Deflection 

If a spring is made of an elliptical bar coiled upon the edge xd, 
as shown, then the length of the bar will be 

7T h D 

1 = - 

x d 
d 

and the value of c will be — when x is unity or less than unity, and 

2 

x d 

-when x is more than unity. 


Then since f 
D 

and since R — — 
2 


SRI 


cG 
















SPRING ENGINEERING. 


69 


we have, 


/irhD \ 1 
\2 )\ xd ) cG 


/ = ■ 


tt S h D 2 


2 x c G d 

Or where x is unity or less, substituting c, 
7 r S h /D 

f = 


x 2 G 


G) 


And when x is more than unity substituting c, 
tt S h / D \ 9 

x G \d) 


f = 


7T 


x 2 G 

which may be expressed as follows: 

t rS 


f = X 2 


When 


Or for steel 


(i) 

lows: 


X, 


where 


f=Y, 


Y, 


Q 


7 tS 


X, = .019946 X 2 


G 


Rectangular Section—Load 

The polar moment of a rectangular section whose outside dimen¬ 
sions are a and b is, 

ab 3 -f- a 3 b 

J — - 

12 

From which the polar moment of a rectangular section whose out¬ 
side dimensions are d and xd, is 

d A 

J — — (X -f- x 3 ) 

12 


In a rectangular section the distance from the neutral axis to the 
remotest fibre will be, always 


1 / d 2 + x 2 d 2 d 

c —-= — 

2 2 


V - 

1 + x 2 














70 


SPRING ENGINEERING. 


Then since 


SJ 


T — 


T — 


Sd 3 (x + x 3 ) PD 



6 )/ l + * ! 


= (j)( XVl + X ')l 


which may be expressed as 


S d 3 

p = — X 3 — 
3 D 


Where 
Or for steel 

Where 


X 3 = x Vl + x 2 
d? 


P = Y 3 


D 


S 

Y 3 = —X 3 — 26666 X 3 
3 


Rectangular Section—Deflection 

If a spring is made of a rectangular section and coiled upon the 
edge xd, as shown, then the length of the bar will be 

7 r h D 

l =- 

x d 


and the value of c will be l/ ef + x 2 ef or d Vl + x° 
SRI D 

Then since / =- and since R = — 

cG 2 

we have 


f 


/ = - 

x j/ 1 + 
which may be expressed as 


*(-)(——)(—=) 

V 2 / \ # d / \d G |/i + a-* / 

1 / \ d / 


where X< = 


-£(*)G)' 


xV 1 + a 2 

Or for steel 

'--Gy* 


where y 4 = 


X 4 = .019946 X 4 


G 
























SPRING ENGINEERING. 


71 


Table of Loads and Deflections for Various Values of x 

From the foregoing we have the following table of co-efficients for 
steel springs, where 

O. D. = a constant 
I. D. =a constant 
d —a constant 

and where the edge upon which the bar is rolled is equal to xd, x being 
a variable. 



Load = P = 

-(v) 

Deflection = / = 

= -(D* 


Y x — Ellip- 

Y 3 — Rec- 

Y 2 — Ellip¬ 

F 4 — Rec¬ 

X 

tical 

Sections 

tangular 

Sections 

tical 

Sections 

tangular 

Sections 

.1 

1,587 

2,680 

.19946 

.198470 

.2 

3,267 

5,439 

.099730 

.097794 

.3 

5,137 

8,352 

.066487 

.063683 

.4 

7,289 

11,488 

.049865 

.046299 

.5 

9,818 

14,907 

.039892 

.035681 

.6 

12,818 

18,659 

.033242 

.028506 

.7 

16,383 

22,786 

.028494 

.023343 

.8 

20,609 

27,320 

.024933 

.019469 

.9 

25,588 

32,289 

.022162 

.016473 

1.0 

31,416 

37,712 

.019946 

.014104 

1.1 

34,715 

43,608 

.016484 

.012197 

1.2 

38,328 

49,986 

.013851 

.010641 

1.3 

42,255 

45,858 

.011802 

.0093549 

1.4 

46,496 

46,232 

.010177 

.0082810 

1.5 

51,051 

72,112 

.0088649 

.0073760 

1.6 

55,920 

80,504 

.0077914 

.0066071 

1.7 

61,104 

89,413 

.0069017 

.0059488 

1.8 

■ 66,602 

98,839 

.0061562 

.0053813 

1.9 

72,414 

108,787 

.0055252 

.0048894 

2.0 

78,540 

119,259 

.0049865 

.0044601 

3.0 

157,080 

252,986 

.0022162 

.0021025 

4.0 

267,036 

439,804 

.0012466 

.0012094 

5.0 

408,408 

679,878 

.00079784 

.00078235 

6.0 

581,196 

973,254 

.00055406 

.00054652 

7.0 

785,400 

1,319,950 

.00040706 

.00040297 

8.0 

1,121,020 

1,719,970 

.00031166 

.00030925 

9.0 

1,288,056 

2,173,321 

.00024625 

.00024474 

10.0 

1,586,508 

2,680,001 

.00019946 

.00019847 















72 


SPRING ENGINEERING. 


SPRINGS MADE OF FLAT MATERIAL 



Fig. 25. 

Front view of R. H. spring, 
wound on edge. 





Fig. 26. 

Sectional view of R. H. spring, 
wound on edge. 



Fig. 27. 

Front view of L. H. Spring, 
wound on flat. 



Fig. 28. 

Showing spring wound L. H. 
and on edge. 






















































MATHEMATICAL TABLES—FOURTH POWERS MEAN DIAMETERS 


No. 

Fourth Power 

No. 

Fourth Power 

No. 

Fourth Power 

No. 

Fourth Power 

* 

0.'000015 

3* 

139.62745 


2153.90261 

io A 

10771.35887 

i 

0.000024 

3# 

150.06250 

6 # 

2234.03929 

10 * 

11038.12865 

A 

0.000124 

3* 

161.07176 

6f| 

2316.39164 

10* 

11309.82394 

* 

0.003906 

3i 

172.67603 

7 

2401.00000 

10# 

11586.50431 

A 

0.009537 

m 

184.89625 

7* 

2487.90532 

10A 

11868.22971 

# 

0.019776 

3 f 

197.75391 

n 

2577.14869 ‘ 

10 i 

12155.06250 

A 

0.036636 

3f| 

211.27077 

7A 

2668.77172 

10* 

12444.75546 

# 

0.062500 

si 

225.46899 

7 i 

2762.81644 

io i 

12744.29331 

A 

0.100113 

3ft 

240.37111 

7A 

2859.32497 

10ft 

13046.81479 


0.152588 

4 

256.00000 

n 

2958.34009 

10# 

13355.79595 

H 

0.223404 

4A 

272.37892 

7* 

3059.90479 

10ft 

13667.98502 

f 

0.316406 

4 i 

289.53149 

7# 

3164.06250 

10* 

13986.75836 

it 

0.435806 

4A 

307.48171 

7 » 

< TS' 

3270.85695 

10ft 

14311.02750 

* 

0.586182 

4 * 

326.25391 

7# 

3380.33231 

11 

14641.00000 

if 

0.772477 

4A 

345.87282 

7f* 

3492.53296 

ii A 

14976.59724 

l 

1.000000 . 

4 f 

366.36352 

7 f 

3607.50394 

iii 

15317.93017 

i* 

1.274429 

4f 7 g 

387.75148 

7f| 

3725.29031 

HA 

15665.06417 


1.601806 

4 i 

410.06250 

7t 

3845.93777 

.m 

16018.06612 

1A 

1.988541 

4A 

433.32277 

7f| 

3969.49224 

11A 

16377,00104 

n. 

2.441406 

4i 

457.55884 

8 

4096.00000 

hi 

16741.93430 

1A 

2.967544 

4H 

482.79762 

8A 

4225.50779 

H A 

17112.93161 

if 

3.574462 

4 f 

509.06641 

8* 

4358.06272 

iii 

17490.06250 

l* 

4.270035 

4ft 

536.39284 

8* 

4493.71216 

H* 

17873.39228 

n 

5.062500 

4 i 

564.80493 

8f 

4632.50395 

Hi 

18262.98892 

l* 

5.960464 

4f| 

594.33107 

8A 

4774.48612 

Ilf# 

18658.91961 

if 

6.972901 

5 T 

625.00000 

8 # 

4919.70724 

11# 

19061.25420 

1H 

8.109146 

5* 

656.84084 

8A 

5068.21632 

lift 

19470.05944 

if 

9.378906 

5# 

689.88306 

8i 

5220.06250 

11* 

19985.40594 

if? 

10.792252 

5* 

724.15651 

8A 

5375.29544 

lift 

20307.36352 

i * 

12.359618 

5* 

759.69141 

81 

5533.96508 

12 

20736.00000 

irf 

14.091811 

5* 

796.51832 

8ff 

5782.99683 

12* 

21171.38711 

a 

16.000000 

5 f 

834.66821 

8 f 

5861.81645 

12* 

21613.59335 

2* 

18.095719 

6* 

874.17238 

8f| 

6031.09990 

12* 

22062.69244 

2 * 

20.390869 

5f 

915.06250 

8t 

6204.02366 

13* 

22518.75360 

2* 

22.897720 

5A 

957.37062 

8H 

6380.63971 

12* 

22981.84886 


25.628906 

5t 

1001.12917 

9 

6561.00000 

12# 

23452.05060 

2A 

28.597427 

5H 

1046.37089 

9A 

6745.15722 

12* 

23929.30971 

2f 

31.816650 

5 f 

1093.12893 

9 # 

6933.16482 

12 i 

24414.06250 

2* 

35.300309 

5ff 

1141.43678 

9* 

7125.07447 

12* 

24906.00804 

2# 

39.062500 

5# 

1191.32839 

9f 

7320.94145 

12# 

25405.37082 

2* 

43.117691 

6ft 

1242.83792 

9A 

7520.81914 

12ff 

25912.19509 

at 

47.480714 

6 

1296.00000 

at 

7724.76197 

12* 

26426.56672 

2H 

52.166764 

6* 

1350.84964 

9* 

7932.82473 

121! 

26948.55687 

2f 

57.191406 

6 # 

1407.42210 

9 # 

8145.06250 

12# 

27478.24215 

21# 

62.570571 

6 A 

1465.75316 

9A 

8361.53080 

13ff 

28015.69826 

2* 

68.061807 

6 i 

1525.87894 

9 f 

£582.28544 

13 

28561.00000 

2ff 

74.458023 

6A 

1587.83571 

9ff 

8807.38258 

13* 

29114.22381 


81.000000 

6| 

1651.66037 

9 f 

9036.87895 

13# 

29675.44519 

3* 

87.985319 

6* 

1717.39013 

9f| 

9270.83136 

13* 

30244.74264 

3 £ 

95.367431 

6# 

1785.06250 

9 # 

9509.29715 

13 # 

30822.19107 

3 A 

103.228775 

6* 

1854.71534 

9ft 

9752.33397 

13* 

31407.86974 

.ID 

8 * 

111.566406 

„ i D 

Of 

1926.38696 

10 

10000.00000 

13# 

32001.85559 

3A 

120.39919 


2000.11596 

10* 

10252.35321 

13* 

32604.22728 

.ID 

3# 

129.74634 

6f 

2075.94137 

10 i 

10509.45334 

13# 

33215.06250 



























MATHEMATICAL TABLES-CUBES OF MEAN DIAMETERS 


No. 

Cube 

No. 

Cube 

No. 

Cube 

| No. 

Cube 

A 

0.000244 

8A 

40.618896 

6*f 

316.169189 

ioa 

1057.311279 

1 

0.001953 

31 

42.875000 

61 

324.951172 

10* 

1076.890625 

A 

0.006592 

»A 

45.213135 

6*1 

333.894287 

ioa 

1096.710205 

* 

0.015625 

3 | 

47.634766 

7 

343.000000 

101 

1116.771484 

A 

0.030518 

3*1 

50.141357 

7A 

352.269775 

10 A 

1137.075928 

£ 

0.052734 

8* 

52.734375 

n 

861.*05078 

101 

1157.625000 

A 

0.083740 

8*| 

55.415283 

7A 

371.307373 

ioa 

1178.201660 

£ 

0.125000 

31 

58.185547 

71 

381.078125 

10f 

1199.462891 

A 

0.177979 

8*1 

61.046631 

7A 

391.018799 

ion 

1220.754639 

1 

0.244141 

4 

64.000000 

7 f 

401.130859 

10* 

1242.306641 

H 

0.324951 

4A 

67.047119 

7 A 

411.415771 

10*1 

1264.091064 

£ 

0.421875 

41 

70.189453 

71 

421.875000 

101 

3286.138672 

18 

1 § 

0.536377 

4A 

73.428467 

7A 

432.510010 

ion 

1308.486768 

£ 

0.669922 

41 

76.765625 

71 

443.322266 

ii 

1331.000000 


0.823975 

4A 

80.202393 

7*1 

454.313232 

li A 

1353.816650 

1 

1.000000 

4 f 

83.740234 

7 f 

465.484375 

ill 

1376.892578 

1A 

1.199463 

4 A 

87.380615 

7*1 

476.837158 

ii A 

1400.229248 

n 

1.423828 

41 

91.125000 

71 

488.373047 

li* 

1423.828125 

1A 

1.674561 

4A 

94.974854 

7*1 

500.093506 

li A 

1447.690673 

i* 

1.953125 

4 1 

98.931641 

8 

512.000000 

ill 

1471.818359 

1A 

2.260986 

4H 

102.996826 

8A 

524.093994 

n A 

1496.212646 

if 

2.599609 

4 f 

107.171875 

81 

536.376953 

ill 

1520.875000 

1A 

2.970459 

41f 

111.458252 

8A 

548.850342 

ii A 

1545.806885 

n 

3.375000 

41 

115.857422 

81 

561.515625 

HI 

1571.009766 

1A 

3.814697 

4*1 

120.370850 

8A 

574.374268 

ii** 

1596.485107 

il 

4.291016 

5 

125.000000 

81 

587.427734 

iif 

1622.234375 

1H 

4.805420 

«A 

129.746338 

8 A 

600.677490 

n*i 

1648.259033 

if 

5.359375 

51 

134.611328 

81 

614.125000 

HI 

1674.560547 

if® 

5.954346 

6A 

139.596436 

8A 

627.771729 

im 

1701.140381 

U 

6.591796 

51 

144.703125 

81 

641.619141 

12 

1728.C00000 

in 

7.273193 

«A 

149.932861 

8*1 

655.668701 

12A 

1755.140869 

2 

8.000000 

51 

155.287109 

8 f 

669.921875 

121 

1782.564453 

2A 

8.773682 

5A 

160.767334 

8*1 

684.380127 

1*A 

1810.272217 

21 

9.595703 

51 

166.375000 

81 

699.044922 

12* 

1838.265625 

2A 

10.467529 

5A 

172.111572 

8*1 

713.917725 

12A 

1866.546143 

21 

11.390625 

51 

177.978516 

9 

729.000000 

12# 

1895.115234 

2A 

12.366455 

5H 

183.977295 

9 A 

744.293213 

12A 

1923.964600 

2 f 

13.396484 

5 f 

190.109375 

91 

759.798828 

12* 

1953.325000 

»A 

14.482178 

5 *f 

1*96.376221 

9 A 

775.518311 

12A 

1982.568604 

21 

15.625000 

51 

202.779297 

91 

791.453125 

121 

2012.306641 

»A 

16.826416 

5*f 

209.320068 

9A 

807.604736 

12 *J 

2042.340576 

21 

18.087891 

6 

216.000000 

91 

823.974609 

12 f 

2072.671875 

2H 

19.410889 

6 A 

222.820557 

9A 

840.564209 

12*1 

2103.302002 

2 f 

20.796875 

61 

229.783203 

91 

857.375000 

121 

2134.232422 

2f | 

22.247314 

6 8 
°TS 

236.889404 

9 A 

874.408447 

12*5 

2165.464600 

21 

23.763672 

61 

244.140625 

91 

890.666016 

33 

2197.000000 

2H 

25.347412 

6A 

251.588330 

9*1 

919.149170 

13A 

2228.840088 

3 

27.000000 

61 

259.083984 

9 f 

926.859375 

13* 

2260.986328 

3A 

28.722900 

6A 

266.779053 

9*| 

944.798096 

13 A 

2293.440180 

3 1 

30.517578 

61 

274.625000 

9* 

962.966797 

13* 

2326.203125 

8A 

32.385498 

6A 

282 623291 

9*| 

981.366943 

13A 

2359,276611 

31 

34.328125 

61 

290.775391 

10 

1000.000000 

13 f 

2392.662109 

8A 

36.346924 

6*1 

299.082764 

ioa 

1018.867432 

13 A 

2426.361084 

31 

38.443359 

6* 

307.546875 

101 

1037.970703 

13* 

2460.375000 
























MATHEMATICAL TABLE-FIFTH POWERS OF BAR DIAMETERS 


No. 

Fifth Power 

No. 

Fifth Power 

No. 

Fifth Power 

No. 

Fifth Power 

A 

0.000000953674 

A 

0.056313 

1* 

1.35408 

lye 

9.3132 

i 

0.0000305176 

1 

0.095367 

1 i 

1.80203 

If 

11.3310 

re 

0.000231743 

if 

0.153590 

1* 

2.36139 

Mi 

13.6842 

i 

0.000976562 

f 

0.237305 

u 

3.05176 

If 

16.4131 

i V 

0.00298023 

1 3 

Til 

0.354093 

l.A 

3.89490 

Iff 

19 5610 

1 

0.00741577 

Y 

0.512909 

if 

4.91489 

If 

23 1743 

A 

0 0160284 

it 

0.724196 

1A 

6.13818 

115 

1 1 6 

27.3029 

i 

0.0312500 

i 

1.000000 

H 

7.59375 

2 

32.0000 



















THE 


WM. D. GIBSON CO. 


SILAS HOWE, President ENOCH PETERSON, Vice-Pres’t. 

WARREN D. HOWE, Treas. ALEXANDER B. PETERSON, Sec’y. 
WILLIAM G. HOWE, Sales Manager 

N. W. Comer Huron and Kingsbury Streets 

CHICAGO 


MANUFACTURERS OF 

SPRINGS 

Compression Special Flat 
Torsion Extension 

MADE FROM 

Crucible Spring Steel Alloy Steel 
Open Hearth Spring Steel 
Music Wire 

Phosphor Bronze and Brass 


Springs For All Purposes 


ALSO 


SPECIAL BENT WIRES AND SHAPES 

FROM ROUND, SQUARE AND FLAT MATERIAL 




THE Wm. D. GIBSON COMPANY 


MANUFACTURERS OF 


Adding Machine Springs 

Agricultural Implement Springs 
Air Brake Springs 
Automobile Springs 

Baby Carriage Springs 
Baby Jumper Springs 
Bed Springs 

Bending Springs 

Bicycle Springs 
Bird Cage Springs 
Bobbin Ring Springs 
Brake Springs 

Buggy Boot Springs 
Car Springs 

Cash Register Springs 
Chair Springs 

Check Rower Springs 
Clutch Springs 
Couch Springs 

Cutter Bar Springs 

Derrick Springs 
Door Check Springs 
Draught Springs 
Drill Springs 

Electrical Equipment Springs 
Engine Springs 
Exerciser Springs 

Fender (Car) Springs 

Folding Cart Springs 
Furniture, Upholstery Springs 
Gas Engine Springs 
Gate Springs 

Go-Cart Springs 
Governor Springs 
Grease Cup Springs 
Gun Springs 

Hay Loader Springs 
Hay Rack Springs 


Helical Springs 
Hinge Springs 
Journal Box Lid Springs 
Lamp Springs 

Lever Springs 

Loader Teeth Springs 
Loom Springs 

Machinery Springs 

Motor Springs 

Organ and Piano Springs 
Oven Door Springs 
Piano Player Springs 

Plow Springs 
Pop Valve Springs 

Pump and Windmill Springs 
Pump Valve Springs 

Rocker Springs 
Sash Springs 

Seat and Back Springs 
Scale Springs 

Shade Roller Springs 
Shuttle Springs 

Squaring Shear Springs 
Switch Spring 

Tedder Fork Springs 
Trace Springs 
Trap Springs 
Trolley Springs 

Typewriter Springs 
Upholstery Springs 
Valve Springs 

Wagon Brake Springs 

Wagon Pole Springs 
Washing Machine Springs 
Window Screen Springs 
Windmill Springs 

Window Shade Springs 
Wringer Springs 


Oil tempered springs of Vanadium Steel. 
Oil tempered springs of Crucible Steel. 
Oil tempered springs of Alloy Steel. 
Springs of Brass and Phosphor Bronze. 


No spring is too large , too small or too odd shaped for us 
to make. We have made compression springs of two-inch 
square steel which weighed Three Hundred pounds each and 
have made piano wire compression springs so small that 
Thirty-Eight Thousand weighed one pound. 






























































































the w./Vf n GiBsojv compass* Chicago ill 

TORSION SPRINGS. 

JTZADE OF ROIL ftD SQUARE AJVD FLAT JYTATER/AL 



The a. bo ve ctr a njings represent ontp cl fexo of l Lie Lcntmn ted 
shapes in zohich torsion, springs can be rnactc. 

We can furnish, ang sice and shape 


































































































WJYl.D. GIBSOJr COJrZPATfY. CSC ICAGO ILL. 


FLAT SPRINGS. 





Otv<s cLe/lecCion and pressure, 
at point A 


si 





Angle 



pressure ctt point A 


Above is rtzerely cl suggestion for drazoi ngs required on orders 
/or /Let t springs 7'urnisb. /ull size drawings zoith all dimensions. 

























































































































































































































































































































































































































































































































































